Vehicle Mechatronic Systems

Abstract

In this chapter, we develop models for various mechatronic components used in modern road vehicles. To start with, we develop a complete vehicle model by integrating its various basic component models like vehicle body, tires, wheels, engine, clutch, gear box, differential, transmission system, suspension, steering, etc. Mechatronic implementations of functionalities of some of these elements are considered next. We consider various active and semi-active suspensions, anti-roll bar, power steering, antilock and regenerative braking systems, and automatic transmission systems. In addition to these, we consider the hybrid vehicle system with power split device (PSD), torque converter, and fuel cells. Detailed models of two types of fuel cells, namely solid oxide fuel cell and proton exchange membrane fuel cell, along with their control circuits are developed at the end of the chapter. This chapter showcases the application of bond graph modeling to chemical kinetics and thermodynamics (engine, fuel cells, heat exchanger, etc.) as part of complex mechatronic systems.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Vehicle, aircraft, and spacecraft industries lead the innovations in mechatronics research. Mechatronization of vehicles has been one of the major developments of our time [25]. Mechatronized vehicles use increasing numbers of mechatronic components (sensors, actuators, controllers) as replacement for mechanical, electrical, hydraulic, and electronic units or act to integrate their functions in a more useful and coordinated manner. Designing such mechatronic components and evaluating their performance in real vehicle applications is a challenging task. These systems need a high degree of reliability and hence the integration of electrical, mechanical, control engineering, and software technologies must be thoroughly designed and tested. This is the reason why hardware-in-the-loop systems (HIL systems) are being used throughout the development process for complex vehicle mechatronic systems. Systematic and automated testing of all electronic control units (ECUs) both as individual components and within the overall system can be performed through the HIL system. Modern automobile systems are using intelligent sensors and actuators. These intelligent units perform various additional tasks such as communication and decision making. The reliability of these intelligent systems needs thorough testing before commercialization. In hardware-in-the-loop simulation (HIL simulation or HILS) the behavior of the vehicle is simulated by software and hardware models. Real vehicle components (ECUs and other electronic/control units) are then connected via their electrical interfaces, to a vehicle dynamics simulator (virtual vehicle), which reproduces the behavior of the real-time environment. One such case has been illustrated in the final chapter of this book.

HILS allows early testing of ECU control algorithms and circuits even before a prototype vehicle has been produced or while the vehicle is being simultaneously developed. This cuts the total development time. Moreover, HILS reduces the number of expensive field trials, allows testing under simulated extreme or hazardous conditions and simulation of faults (e.g., brake failure) and environmental effects (e.g., skidding on snow), and maintains repeatability/reproducibility of tests. Software-in-the-loop simulation (SILS) is a precursor to HILS. It is a virtual prototyping step, i.e., it allows one to test the control logic by plugging in the control software code to the vehicle model. At this time, it is important to note that the accuracy of the SILS/HILS, and consequently the prototype development, is highly dependent on the accuracy of the vehicle model along with the control architecture. SIMPACK Automotive and Saber are two well-known software suites for HIL and SIL simulation. This chapter is devoted to modeling and simulation of complex vehicle mechatronic systems individually and integrated with the full vehicle dynamics model.

Some of the vehicle mechatronic systems are so simple that we often overlook them, e.g., speed, fuel and other indicators, central locking system, automatic door closure system, rear and side view mirror positioning systems, headlight positioning system, warning buzzer, heating/air-conditioning regulation system, electronic massage system at lumbar support, power seat adjuster, power window regulator, communication with traffic routing systems, etc. And then there are complex mechatronic systems like multi-point fuel injection systems, air–fuel ratio control system, microprocessor controlled catalytic converter system for emission control, power assisted steerings, active and semi-active suspension systems, automatic transmission system, hybrid drive systems and power trains, anti-lock and regenerative brakes, cruise control or automatic navigation system, self-diagnostic (early troubleshooting and warning) system, voice command system, driver alarm at low coefficient of friction and risk of aquaplaning, counter-steering by active bearing kinematics in case the friction coefficient is different on wheels on opposite sides, fault accommodation (load, side wind, brake squeal, crossfall of road), etc.

Most of these vehicle mechatronic systems are simple electronic or electromechanical systems which do not require much of control action. On the other hand, critical vehicle mechatronic systems use microprocessors to carry out complex processing. Bond graph modeling allows easy integration of vehicle model with mechatronic systems [31, 45–47]. We will discuss a few of those complex systems in this chapter.

1 Model of a Four Wheel Vehicle

A road vehicle constitutes a multibody system. The vehicle body, each wheel, each axle, gear box, differential, etc. can be each considered as a 6 DOF rigid body. These rigid bodies are constrained by means of various joints. In the present case, we reduce the degrees of freedom of the system by neglecting trivial dynamics. We assume that the engine, differential, etc. are rigidly mounted on the vehicle body. Wheels are rigidly fixed to the axles which are attached to the body through suspension systems.

Fig. 6.1

Four wheel vehicle model

The schematic representation of the full vehicle model¹ is shown in Fig. 6.1.

1.1 Word Bond Graph Representation

The word bond graph representation of the full vehicle model is shown in Fig. 6.2 where bonds represented by two parallel lines are multi-bonds. In the word bond graph, the global system is decomposed into six subsystems. These are: vehicle body, suspension, wheel, steering, antilock braking system, and differential. The flow variables at the interface of different subsystems are marked in Fig. 6.2. The complementary power variables or generalized effort variables (force for linear velocity and torque for angular velocity) are not shown in Fig. 6.2 to maintain clarity of the figure. The four wheels are connected to the vehicle body through suspensions. The steering and ABS (a brake system) are coupled with the axle by scalar bonds.

Likewise, scalar bonds connect the differential to the rear wheels and the vehicle body. The model architecture and its submodels are developed in line with those presented in [19, 28, 57].

1.2 Tire Slip Forces and Moments

The tire forces and moments from the road surface act on the tire as shown in Fig. 6.3. The tire characteristics play an important role in the dynamic behavior of vehicles. The forces acting along x, y, and z axes are longitudinal force $F_{\mathrm{x}}$, lateral force $F_{\mathrm{y}}$ and normal force $F_{\mathrm{z}}$, respectively. Similarly, the moments acting along x, y, and z axes are overturning moment $M_{\mathrm{x}}$, rolling resistance moment $M_{\mathrm{y}}$ and the self-aligning moment $M_{\mathrm{z}}$, respectively [62].

Fig. 6.2

Word bond graph of four wheel vehicle model

Fig. 6.3

Tire forces and moments

The longitudinal slip arises due to application of torque about the wheel spin-axis. In actual case, as the vehicle speed cannot be measured, the longitudinal slip is calculated by measuring the wheel speed and acceleration [56]. The longitudinal slip ratio is defined as the normalized difference between the circumferential velocity and the translational velocity of the driven wheel [40]. It is expressed as:

$$\begin{aligned} \sigma _{\mathrm{x}}=\left\{ \begin{array}{l} \dfrac{\dot{\theta }_{\mathrm{wy}}r_{\mathrm{w}}-\dot{x}_{\mathrm{w}}}{\dot{\theta } _{\mathrm{wy}}r_{\mathrm{w}}}\left( \mathrm{during }\;\mathrm{traction,\; assuming}\; \dot{\theta }_{\mathrm{wy}}\rangle 0\right) \\ \dfrac{\dot{x}_{\mathrm{w}}-\dot{\theta }_{\mathrm{wy}}r_{\mathrm{w}}}{\dot{x}_{ \mathrm{w}}}\left( \mathrm{during }\;\mathrm{braking,\; assuming}\dot{x}_{\mathrm{w} }\rangle 0\right) \end{array} \right. \end{aligned}$$

(6.1)

Lateral wheel slip is the ratio of lateral velocity to the forward velocity of wheel [58]. It is given as:

$$\begin{aligned} \sigma _{\mathrm{y}}=\tan \alpha =\frac{\dot{y}_{\mathrm{w}}}{\dot{x}_{\mathrm{w}}} \end{aligned}$$

(6.2)

If the longitudinal slip ratio $\sigma _{\mathrm{x}}$ and lateral slip ratio $\sigma _{\mathrm{y}}$ are known then for small slip ratios, the longitudinal force $F_{\mathrm{x}}$ and lateral force $F_{\mathrm{y}}$ can be, respectively, approximated as $F_{\mathrm{x}}=\sigma _{\mathrm{x}}C_{\mathrm{x}}$ and $F_{\mathrm{y}}=\sigma _{\mathrm{y}}C_{\mathrm{y}}$ where, $C_{\mathrm{x}}$ and $C_{\mathrm{y}}$ are longitudinal tire stiffness (coefficient) and cornering coefficient, respectively. However, these linear relations are invalid for large slip ratios.

The empirical magic formula based on experimental data is adopted for the development of tire–road friction model. It gives more accurate results for larger slip angles. It is also applicable to a wide range of operating conditions. Longitudinal slip velocity develops longitudinal force $F_{\mathrm{x}}$, whereas side slip velocity and camber angle $\gamma$ generate side force $F_{\mathrm{y}}$ and self-aligning moment $M_{\mathrm{z}}$. Pacejka’s magic formula states that longitudinal force, side force, and the self-aligning moment are functions of longitudinal slip and side slip, respectively. Though this formula does not consider the velocity dependence of friction, it is used here for its simplicity. It is given as:

$$\begin{aligned} y_{ \mathrm{o}}=D\sin \left[ C\tan ^{-1}\left\{ Bx_{\mathrm{i}}-E\left( Bx_{\mathrm{i} }-\tan ^{-1}\left( Bx_{\mathrm{i}}\right) \right) \right\} \right] \end{aligned}$$

(6.3)

where output variable, $y_{\mathrm{o}}:F_{\mathrm{x}}, \; F_{\mathrm{y}}$ or $M_{\mathrm{z}}$ and input variable, $x_{\mathrm{i}}:\sigma _{\mathrm{x}}$ or $\sigma _{\mathrm{y}}$.

Ply-steer, rolling resistance, and conicity effects may cause slight variation in the function in Eq. 6.3, but these variations may be neglected. The constant parameters ($B$, $C$, $D$, $E$) can be determined by measuring the tire forces and moments by sophisticated equipments. This formula is unsuitable when snow and ice start significantly affecting the vehicle performance.

The tire friction models are generally nonlinear in nature. The coefficient of friction (Fig. 6.4) is

$$\begin{aligned} \mu = \frac{\sqrt{F_{\mathrm{x}}^{2}+F_{\mathrm{y}}^{2}}}{F_{\mathrm{z}}} \end{aligned}$$

where longitudinal force $F_{\mathrm{x}}$ and side force $F_{\mathrm{y}}$ can be found out under particular conditions of constant value of linear and angular velocity from the most widely used static magic formula (Eq. 6.3) given by Pacejka [58] and $F_{\mathrm{z}}$ is the normal force. However, one main disadvantage of this model is its inability to describe low slip effects and large forward and side slip effects during wheel lockup.

Another friction model often used to model tire forces is given by Burckhardt as follows [55]:

$$\begin{aligned} \mu (\sigma _{\mathrm{x}},\dot{x}_{\mathrm{c}})=\left[ C_{1}\left( 1-\mathrm{e}^{-C_{2}\sigma _{\mathrm{x}}}\right) -C_{3}\sigma _{\mathrm{x}}\right] \mathrm{e}^{-C_{4}\sigma _{\mathrm{x}}\dot{x} _{\mathrm{c}}} \end{aligned}$$

(6.4)

Fig. 6.4

Friction coefficient versus slip ratio curves for different road surfaces

where constant parameters $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$ are determined from experiments. These parameters for asphalt dry road conditions are given in [55]. The normal load at the tire is kept constant for using this model. This model considers the velocity dependence of friction force. The main drawback of this model is the nonlinearity of its parameters which are difficult to estimate.

1.3 Vehicle Body

The vehicle body is considered a 6 DOF rigid body. Wheel axles and wheels are attached to the body. Wheel axles are attached to the body through suspension systems. The vehicle body motion is described by three linear displacements along body-fixed x, y, and z axes and the rotational motion of the body can be defined by the three Cardan angles. The Newton–Euler equations of the vehicle body with attached body fixed axes aligned with the principal axes of inertia are as follows:

$$\begin{aligned} \sum F_{ \mathrm{x}}&=m_{\mathrm{c}}\ddot{x}_{\mathrm{c}}+m_{\mathrm{c}}(\dot{z}_{\mathrm{c}} \dot{\theta }_{\mathrm{cy}}-\dot{y}_{\mathrm{c}}\dot{\theta }_{\mathrm{cz}}), \end{aligned}$$

(6.5)

$$\begin{aligned} \sum F_{\mathrm{y}}&=m_{\mathrm{c}}\ddot{y}_{\mathrm{c}}+m_{\mathrm{c}}(\dot{x}_{ \mathrm{c}}\dot{\theta }_{\mathrm{cz}}-\dot{z}_{\mathrm{c}}\dot{\theta }_{\mathrm{cx} }), \end{aligned}$$

(6.6)

$$\begin{aligned} \sum F_{\mathrm{z}}&=m_{\mathrm{c}}\ddot{z}_{\mathrm{c}}+m_{\mathrm{c}}(\dot{y}_{ \mathrm{c}}\dot{\theta }_{\mathrm{cx}}-\dot{x}_{\mathrm{c}}\dot{\theta }_{\mathrm{cy} }), \end{aligned}$$

(6.7)

$$\begin{aligned} \sum M_{\mathrm{x}}&=J_{\mathrm{cx}}\ddot{\theta }_{\mathrm{cx}}+\dot{\theta }_{ \mathrm{cz}}\dot{\theta }_{\mathrm{cy}}(J_{\mathrm{cz}}-J_{\mathrm{cy}}), \end{aligned}$$

(6.8)

$$\begin{aligned} \sum M_{\mathrm{y}}&=J_{\mathrm{cy}}\ddot{\theta }_{\mathrm{cy}}+\dot{\theta }_{ \mathrm{cx}}\dot{\theta }_{\mathrm{cz}}(J_{\mathrm{cx}}-J_{\mathrm{cz}}), \end{aligned}$$

(6.9)

$$\begin{aligned} \sum M_{\mathrm{z}}&=J_{\mathrm{cz}}\ddot{\theta }_{\mathrm{cz}}+\dot{\theta }_{ \mathrm{cy}}\dot{\theta }_{\mathrm{cx}}(J_{\mathrm{cy}}-J_{\mathrm{cx}}). \end{aligned}$$

(6.10)

The equations for three linear velocities of left-front suspension reference point (point 1 in Fig. 6.1) in the moving system of axes are

$$\begin{aligned} \dot{x}_{1}&=\dot{x}_{c}+z_{1}\dot{\theta }_{\mathrm{cy}}-y_{1}\dot{\theta }_{ \mathrm{cz}}, \\ \nonumber \dot{y}_{1}&=\dot{y}_{c}+x_{1}\dot{\theta }_{\mathrm{cz}}-z_{1}\dot{\theta }_{ \mathrm{cx}}, \\ \nonumber \dot{z}_{1}&=\dot{z}_{c}+y_{1}\dot{\theta }_{\mathrm{cx}}-x_{1}\dot{\theta }_{ \mathrm{cy}}. \end{aligned}$$

(6.11)

The equations for the three angular velocities of left front suspension reference point are similarly expressed. Likewise, the equations for three linear velocities and three angular velocities of other suspension reference points can be derived.

It is assumed that the aerodynamic effects, although considered in the model, are not very significant and that the vehicle is symmetrical with respect to its longitudinal axis. The bond graph in Fig. 6.5 models the vehicle body inertia and transformations of the three linear and three angular velocities into velocities at relevant suspension reference points. The vehicle body is modeled as a rigid body with six degrees of freedom, i.e., pitch, roll, yaw, heave, surge, and sway motions. The rigid body motion is described with respect to a coordinate system rotating and translating with it. This local coordinate frame attached at the center of mass of the body is assumed to be aligned with the inertial principal axes. The inertias are coupled by a pair of gyrator rings (Euler junction structure), one for translational and the other for rotational velocities, according to Eqs. 6.5–6.10.

Fig. 6.5

Bond graph model of vehicle body

Three sets of forces and moments act on the body. First, the weight of the body and the aerodynamic forces (modeled by R$_{ \mathrm{aero}}$ element) in the inertial frame act on the vehicle body model in non-inertial frame through coordinate transformation (CTF) [21]. Second, engine torque which is in a different body-fixed frame (wheel frame) is transformed twice to act on the body (through the differential). Lastly, the suspension forces and moments (constraint forces) acting in the inertial frame are transformed to get forces in the body-fixed direction. Then these forces are multiplied with the moment arms. The forces at all the four corners of the body are added together to get the suspension force acting at the center of gravity of the body. For display purposes, the angles of rotation in the inertial frame are obtained by integration of the angular velocities in the body-fixed frame after CTF.

1.4 Suspension System

Depending on the desired behavior of the vehicle, various types and forms of suspensions are used. Single wheel suspension system is the only option when an independent suspension system is required. Suspensions which can damp the vertical motion only slightly, due to the axle stiffness or axle fixed in a hinge point, are some typical single wheel suspension system arrangements. In real vehicles, the wheels are not connected to the car body only by an axle but through suspension systems which form a link between the vehicle body and the wheels. Suspension systems contribute to a car’s smooth running and safe acceleration or braking. There are several suspension systems, e.g., McPherson, pseudo-McPherson, trailing arms, multi-arms, etc. [49]. A suspension system can be approximated by simple linear spring-damper suspension system which corresponds to energy storage and energy dissipation at each corner of the vehicle body while allowing free rotation of the wheel about the axle ($y$-axis).

Fig. 6.6

Bond graph model of suspension system (Linear motion is on left and angular motion on right)

The suspension system is used for the attachment of car body and wheel in a vehicle. The forces are calculated in the suspension for connecting two body components. In the bond graph model [6] given in Fig. 6.6, the C- and R-elements, which are in parallel, model the energy storage and dissipation due to relative motion between the car body and the wheel. The values of the stiffness and damping in the z-direction are the actual suspension stiffness and damping parameters, whereas much higher values of stiffness and damping in the longitudinal and lateral directions are considered to implement the constraints which limit the relative motions in x and y directions.

1.5 Wheels

The wheels are modeled by their mass, rotary inertia, radius, and tire stiffness. The tire is the most important of the wheel components. So tire forces and moments play an important role in vehicle dynamics. Tire forces are necessary to control the vehicle. As the tires are the only means of contact between the road and the vehicle, they are the key factors determining the vehicle handling performance. Tire models are broadly classified as physical, analytical, and empirical models. The physical models are constructed to predict tire elastic deformation and tire forces. In such models, complex numerical methods are required to solve the equations of motion. Analytical models are not useful at large slip and at combined slip. Empirical models based on experimental correlations are generally more accurate. Here, both longitudinal and cornering forces and self-aligning moment are modeled empirically as per Pacejka’s magic formula [58].

The bond graph model of wheel is shown in Fig. 6.7. The wheel is modeled as a rigid body with six degrees of freedom. Similar to vehicle body, the inertias are coupled by a pair of gyrator rings, one for the translational and the other for the rotational velocities.

Fig. 6.7

Bond graph model of a wheel

The wheel vertical dynamics is decoupled from longitudinal and cornering dynamics. The suspension force ($F_{ \mathrm{z}}$) and wheel radius ($r_{\mathrm{w}}$) are used to modulate longitudinal and cornering dynamics. $K_{\mathrm{t}}$ and $R_{\mathrm{t}}$ represent the vertical stiffness and damping of tire. The MR-element which is used for modeling road–tire interactions is modulated by the normal reaction, i.e., the suspension force ($F_{ \mathrm{z}}$). Kinetic phenomenon and weight of the wheel are considered for the unsprung mass vertical dynamics. The tire–road interactions and braking action are considered for wheel longitudinal and cornering dynamics. The cornering force and self-aligning moment are dependent on vertical load and lateral slip angle, while the longitudinal force is dependent on vertical load and longitudinal slip rate. The characteristic relations for the MR-element are given according to Pacejka’s magic formulae (Eq. 6.3) or the composite slip based formulation [68].

In Fig. 6.7, the numbers shown within the circles identify the ports for interfacing this submodel to other submodels. Ports 1–6 are connected to the corresponding velocities of the suspension reference point. The braking torque is applied on the front wheel through port 7 and the engine torque is transmitted to the rear wheel through the same port (port 7). Port 8 (shown as a dotted bond) is present only for front wheels. This port is used to interface the front wheel model to the anti-roll bar model, which constrains the relative roll between two front wheel axles.

1.6 Steering System

One basic feature of a good vehicle is its directional stability, which is the ability to steer smoothly. The steering action is achieved by rotating the front wheels of the vehicle around the vertical axis. Vehicles do not steer both the front wheels by an equal amount. While negotiating a curve the inner tires of the car will follow a circle with smaller radius than the outer tires. This distinction comes from the Ackermann steering concept. If a and b are the distances of front and rear wheels from vehicle c.g., c is half track width, $\theta$ is wheel angle, $\delta$ is steering angle and subscripts st, I and r refer to steering, left and right wheels respectively, then the rates of change of wheel steering angles in terms of steering input at the center are

$$\begin{aligned} \dot{\theta }_{\mathrm{1}}=\left[ \dfrac{\left( a+b\right) \cos ^{2}\theta _{ \mathrm{l}}+c\tan \theta _{\mathrm{l}}\cos ^{2}\theta _{\mathrm{l}}}{\left( a+b\right) \cos ^{2}\theta _{\mathrm{st}}-c\tan \theta _{\mathrm{st}}\cos ^{2}\theta _{\mathrm{st}}}\right] \dot{\theta }_{\mathrm{st}} \end{aligned}$$

(6.12)

$$\begin{aligned} \dot{\theta }_{\mathrm{r}}=\left[ \dfrac{\left( a+b\right) \cos ^{2}\theta _{ \mathrm{r}}-c\tan \theta _{\mathrm{r}}\cos ^{2}\theta _{\mathrm{r}}}{\left( a+b\right) \cos ^{2}\theta _{\mathrm{st}}+c\tan \theta _{\mathrm{st}}\cos ^{2}\theta _{\mathrm{st}}}\right] \dot{\theta }_{\mathrm{st}} \end{aligned}$$

(6.13)

We are directly modeling the input from the human operator. The moment applied on the steering wheel about the z-axis is supplied as shown in Fig. 6.8a. The actual rate of rotation of the steering wheel is measured by a yaw rate sensor. It is assumed that the human vision system prescribes the desired rate of rotation. A feedback yaw rate controller may be developed where the controller action can mimic the driver’s response to the consequences of steering. The steering model output is used to rotate the front wheels by applying torques on the front axle (and reactive torques on the vehicle body). The rotary inertia $J_{ \mathrm{st}}$ of the steering wheel is represented by I-element. The remainder of the system comprises mechanical transmission to the front axle. The rate of rotations of left and right wheels about the z-axis are represented at $1_{\dot{\theta }_{\mathrm{l}}}$ and $1_{\dot{\theta }_{\mathrm{r}}}$ junctions. These are interfaced to the front axle model through ports 9 and 10. The modulated transformer (MTF) moduli are determined from Ackermann’s formulae given in Eqs. 6.12–6.13 (see ref. [8] for details).

Fig. 6.8

Bond graph model of steering system with (a) torque input and (b) angular velocity input

In experimental investigations, steering wheel rotations (driver inputs) are measured. The model in Fig. 6.8a may be modified to the form shown in Fig. 6.8b to take the rate of steering wheel rotation as the input. The transformer with modulus $\mu _{\mathrm{stw}}$ is used to scale steering wheel rotation to steering rotation.

1.7 Slider-Crank System

An internal combustion engine with crank shaft and connecting rod is shown in Fig. 6.9. The thermodynamic process of the engine transforms heat into mechanical power.

Fig. 6.9

An internal combustion engine with crank shaft and connecting rod

The inertia of the slider, crank, and piston are negligible with respect to that of the rest of the system, i.e., flywheel, transmission shaft, gears, vehicle mass, and tires. Thus, a simple kinematic model may be developed by neglecting the inertias of the slider, crank, and piston. Note that it is possible to develop a full model of the slider-crank system along the lines of the model developed for the seven-body mechanism (Andrew’s mechanism).

For the simplified model, the relationship between angular velocity of crank shaft and linear velocity of piston as well as the relationship between torque and force are obtained from the kinematic relations. The velocity of the piston is given as

$$\begin{aligned} \dot{x}_{\mathrm{P}}=r\sin \theta _{1}\dot{\theta }_{1}+l\sin \theta _{2}\dot{ \theta }_{2}, \end{aligned}$$

(6.14)

where $\theta _{1}$ (angular position of the crank) and $\theta _{2}$ are the angles shown in the figure, $r$ is the crank radius, and $l$ is the length of the connecting rod. From kinematic analysis, it follows that

$$\begin{aligned} r\sin \theta _{1}=l\sin \theta _{2}\mathrm{\ \ and\ \ }\dot{\theta }_{2}=\dfrac{ \left( r/l\right) \cos \theta _{1}}{\left[ 1-\left( r/l\right) ^{2}\sin ^{2}\theta _{1}\right] ^{1/2}}\dot{\theta }_{1}. \end{aligned}$$

(6.15)

Substitution of $\theta _{2}$ and $\dot{\theta }_{2}$ from Eq. 6.15 into Eq. 6.14 yields the angular velocity of crank shaft $\dot{\theta }_{1}$ in terms of velocity of piston as

$$\begin{aligned} \dot{\theta }_{1}=\dfrac{1}{r}\left[ \dfrac{\left( r/l\right) \sin 2\theta _{1}}{2\left[ 1-\left( r/l\right) ^{2}\sin ^{2}\theta _{1}\right] ^{1/2}} +\sin \theta _{1}\right] ^{-1}\dot{x}_{\mathrm{P}}. \end{aligned}$$

(6.16)

This relation between two flow variables ($\dot{x}_{\mathrm{P}}$ and $\dot{ \theta }_{1}$) is represented by a MTF element in bond graph model, with the angular position of the crank shaft as the modulating variable. The transformer modulus is

$$\begin{aligned} \mu _{\mathrm{T}}=\dfrac{1}{r}\left[ \dfrac{\left( r/l\right) \sin 2\theta _{1} }{2\left[ 1-\left( r/l\right) ^{2}\sin ^{2}\theta _{1}\right] ^{1/2}}+\sin \theta _{1}\right] ^{-1}. \end{aligned}$$

(6.17)

1.8 Engine

The engine model comprises different energy domains: thermal, hydraulic, mechanical, and chemical. In the following, each energy domain is separately considered to develop the engine model in a step-by-step manner.

1.8.1 Engine Thermodynamics

We will first derive the thermodynamic relations for a collapsible chamber . It is convenient to consider pseudo-power variables in thermodynamic formulations [20, 32, 39, 48]. In a pseudo-bond graph, the product of the effort and flow variables does not yield a power variable. However, all rules for true bond graphs are applicable to pseudo-bond graphs and the energetic consistency is maintained by including additional sources (this part is tricky and should be properly handled). On the other hand, true bond graphs of thermodynamic systems are inherently energetically consistent (and preferred) but may lead to very complex model structure and constitutive relations [70–73] (See the section on solid oxide fuel cell modeling at the end of this chapter).

The control volume contains mass $m$ in a variable volume $V$ and with energy $E$. At this time, we assume that ideal gas law is applicable. Note that this approximation can represent the real case with the use of virial coefficients. The gas pressure, temperature, and enthalpy are denoted by $P$, $T$, and $H$, respectively.

When heat $\delta Q$ is introduced into a thermodynamic system containing unit mass,

$$\begin{aligned} dE\,\mathrm{or}\,dU=\delta Q-pdV, \end{aligned}$$

(6.18)

where $U$ is the internal energy. From definition of specific heat at constant volume, $C_{\mathrm{v}}=\left. \dfrac{\partial u}{dT}\right|_{V}=\left. \dfrac{\delta Q}{dT}\right|_{V}$ where $u$ is the specific internal energy. The temperature of the gas can be written as ($U$ and $E$ are given the same status)

$$\begin{aligned} T=\dfrac{E}{mC_{\mathrm{v}}}. \end{aligned}$$

(6.19)

From ideal gas law,

$$\begin{aligned} PV=mRT \end{aligned}$$

(6.20)

which can be written in the form

$$\begin{aligned} P=\dfrac{RE}{VC_{\mathrm{v}}}, \end{aligned}$$

(6.21)

where $R$ is the characteristic gas constant.

We define $\dot{m}$, $\dot{V}$ and $\dot{E}$ as the flow variables and $P$, $P$, and $T$ as the corresponding effort variables. Obviously, the product of complementary power variables (effort times flow) is not power. Thus, we have pseudo-power variables with which we can construct a pseudo-bond graph.

Fig. 6.10

Pseudo-bond graph model of a collapsible gas chamber with mass and heat transfer ports

The pseudo-bond graph given in Fig. 6.10 is a three-port $C$-field in which the power variables are labeled in the bonds. This $C$-field was introduced initially in [34] and later used in [7, 26] to simulate an engine. In integral causality, this C-field receives information of three flow variables ( $\dot{m}$, $\dot{V}$ and $\dot{E}$) and computes the effort variables as follows:

$$\begin{aligned} T\left( t\right)&=\dfrac{\mathop {\displaystyle \int }\limits _{0}^{t}\dot{E}\left( \tau \right) d\tau +E_{0}}{\left( \mathop {\displaystyle \int }\limits _{0}^{t}\dot{m}\left( \tau \right) d\tau +m_{0}\right) C_{\mathrm{v}}}, \end{aligned}$$

(6.22)

$$\begin{aligned} P\left( t\right)&=\dfrac{R\left( \mathop {\displaystyle \int }\limits _{0}^{t}\dot{E}\left( \tau \right) d\tau +E_{0}\right) }{\left( \mathop {\displaystyle \int }\limits _{0}^{t}\dot{V}\left( \tau \right) d\tau +V_{0}\right) C_{\mathrm{v}}}. \end{aligned}$$

(6.23)

The state variables are time integrals of flow variables $m$, $V$, and $E$,for which corresponding initial values at $t=0$ are prescribed as $m_{0}$, $V_{0}$, and $E_{0}$, respectively. For a given position of the crank shaft, the volume of the chamber is known. For example, if at $t=0$ the piston is at top dead center with $\theta =0$, then the initial condition may be prescribed as $V_{0}=V_{c}$, where $V_{c}$ is the clearance volume. At start-up, the pressure and temperature inside the chamber are the same as that of the atmosphere, i.e., $P=P_{0}$ and $T=T_{0}$. Then, the other initial values for state variables are obtained from back calculation as

$$\begin{aligned} m_{0}=\dfrac{P_{0}V_{0}}{RT_{0}}\quad \mathrm{and}\quad E_{0}=m_{0}C_{V}T_{0}. \end{aligned}$$

(6.24)

Note that it is possible to create a true bond graph model of the collapsible chamber. Although the true bond graph model is slightly complicated, it gives the right physics of the system (the growth of entropy, irreversibility of the system, entropy of mixing, and so on). The true bond graph model of the collapsible chamber is given at the end of this chapter where it is used to model the cathode and anode channels of a solid oxide fuel cell.

1.8.2 Mechanical Work

The mechanical work output from the engine is used to rotate the crank shaft which is in turn connected through the transmission system to the end load (wheels in contact with ground). The force balance equation for the piston is

$$\begin{aligned} P_{0}A_{\mathrm{P}}-PA_{\mathrm{P}}-R_{\mathrm{P}}\dot{x}_{\mathrm{P}}-\mu _{\mathrm{T} }\tau _{\mathrm{c}}=0, \end{aligned}$$

(6.25)

where $A_{\mathrm{P}}$ is the cross-sectional area of the piston, $P_{0}$ is the atmospheric pressure acting on the free side of the piston, $R_{\mathrm{P} }$ is the resistance between the piston and the cylinder, $\mu _{\mathrm{T}}$ is the transformer modulus defined in Eq. 6.17, and $\tau _{\mathrm{c}}$ is the load torque. The bond graph model of mechanical coupling with the engine is given in Fig. 6.11. Note that the left side TF-element scales effort variable pressure $P$ to force $F_{\mathrm{P}}=-PA_{\mathrm{P}}$ (positive force is in upward direction) and at the same time, it scales flow variable $\dot{x}_{\mathrm{P}}$ to another flow variable $\dot{V}=-A_{\mathrm{P}}\dot{x}_{\mathrm{P}}$ (positive piston velocity reduces gas volume).

Fig. 6.11

Bond graph model of the piston and crank shaft connected to the engine

The heat generated due to friction should be ideally added to the metallic body. This calls for the use of RS-element (the name suggests resistance $+$ source [69]) in place of R-element. The two-port RS-element shown in Fig. 6.11 receives two information: flow variable $\dot{x} _{\mathrm{P}}$ and body temperature $T_{\mathrm{b}}$. The resistance between the piston and the cylinder depends upon many factors such as the body temperature, level of lubrication, age of the piston rings and engine unbalance forces. If temperature is too high, then the resistance increases and engine seizure may take place. The RS-element uses the two input information to compute two outputs: the resistance force ($R_{ \mathrm{P}}\dot{x}_{\mathrm{P}}$) and the rate of heat generation ($R_{\mathrm{P}} \dot{x}_{\mathrm{P}}^{2}$) due to the resistance.

1.8.3 Mass and Heat Transfer

The mass transfer into the chamber takes place during the suction phase and mass transfer out of the chamber takes place during the discharge phase. These phases are governed by the cam position which in turn depends on the angular position of the crank shaft. A typical valve timing diagram in shown in Fig. 6.12 in which TDC and BDC refer to top and bottom dead centers, respectively. The timings may change for different engines. In four stroke engines, intake, compression, combustion, and exhaust strokes complete during two crankshaft rotations, i.e., two crankshaft rotations are needed per working cycle of the engine. This is the commonly used cycle in vehicles. The exhaust and intake durations are comparatively large in a four-stroke cycle and thus the engine efficiency increases and the pollution level falls. Consequently, the valve timing diagram is different for the four-stroke engine.

Fig. 6.12

Valve timing diagram for a two-stroke engine

For the flows to take place, it is not just enough to have the valve open. The pressure difference must be so that the flow takes place in the desired direction (into or out of the chamber). Nozzle flow equation [26, 34] is used to model flow through the valves.

The critical pressure ratio in Nozzle flow equation is given as

$$\begin{aligned} P_{ \mathrm{cr}}=\left( \dfrac{2}{\gamma -1}\right) ^{\gamma /\left( \gamma -1\right) }, \end{aligned}$$

(6.26)

where $\gamma$ is the ratio of specific heat at constant pressure and specific heat at constant volume. This critical pressure determines the flow choking condition (or the so-called von misses asymptote). The pressure ratio is considered as

$$\begin{aligned} P_{\mathrm{r}}=\left\{ \begin{array}{ll} P_{\mathrm{d}}/P_{\mathrm{u}}&\mathrm{if}\,P_{\mathrm{d}}/P_{\mathrm{u}}>P_{\mathrm{cr}}, \\ P_{\mathrm{cr}}&\mathrm{if}\,P_{\mathrm{d}}/P_{\mathrm{u}}<P_{\mathrm{cr}}, \end{array} \right. \end{aligned}$$

(6.27)

where $P_{\mathrm{u}}$ and $P_{\mathrm{d}}$ refer to pressures in the upstream and downstream sides, respectively.

The mass flow rate of air fuel mixture through the suction valve is given as

$$\begin{aligned} \dot{m}_{\mathrm{i}}=\dfrac{A_{\mathrm{v}_{\mathrm{i}}}C_{\mathrm{d}_{\mathrm{i}}}P_{ {0}}}{\sqrt{T_{\mathrm{0}}}}\sqrt{\dfrac{2\gamma }{R\left( \gamma -1\right) }\left[ P_{\mathrm{r}}^{2/\gamma }-P_{\mathrm{r}}^{\left( \gamma -1\right) /\gamma }\right] }, \end{aligned}$$

(6.28)

where $P_{\mathrm{r}}=P/P_{\mathrm{0}}$ or $P_{\mathrm{cr}}$ (as defined in Eq. 6.27), $A_{\mathrm{v}_{\mathrm{i}}}$is the full valve opening area of the inlet valve and $C_{\mathrm{d}_{\mathrm{i}}}$is the coefficient of discharge of the inlet valve (a function of the cam position or crank shaft position $\theta$). The fuel mass flow rate is $\dot{m}_{ \mathrm{f}}=\alpha \dot{m}_{\mathrm{i}}$ where $\alpha$ the fuel-air ratio, and $\dot{m}_{\mathrm{i}}=\dot{m}_{\mathrm{f}}+\dot{m}_{\mathrm{a}}$ where $\dot{m}_{ \mathrm{a}}$ is the air mass flow rate. The fuel air ratio $\alpha$ is controlled by the throttle valve position, i.e., the pressure on the accelerator or gas pedal. It is necessary to separately keep track of the fuel intake to model the combustion phenomenon.

The discharge mass flow rate is governed by a similar equation:

$$\begin{aligned} \dot{m}_{\mathrm{o}}=\dfrac{A_{\mathrm{v}_{\mathrm{o}}}C_{\mathrm{d}_{\mathrm{o}}}P}{ \sqrt{T}}\sqrt{\dfrac{2\gamma }{R\left( \gamma -1\right) }\left[ P_{r}^{2/\gamma }-P_{r}^{\left( \gamma -1\right) /\gamma }\right] }, \end{aligned}$$

(6.29)

where the chamber is the upstream side and atmosphere is the downstream side ($P_{\mathrm{r}}=P_{\mathrm{0}}/P$ or $P_{\mathrm{cr}}$), $A_{\mathrm{v}_{\mathrm{o} }}$is the full valve opening area of the exhaust valve and $C_{\mathrm{d}_{ \mathrm{o}}}$is the coefficient of discharge of the exhaust valve which is again a function of the cam position.

From definition of enthalpy,

$$\begin{aligned} H=U+PV, \end{aligned}$$

(6.30)

which gives

$$\begin{aligned} dH=dU+PdV+Vdp. \end{aligned}$$

(6.31)

For unit mass of gas at constant pressure, $dH=dU+PdV=\delta Q$, from which the specific heat at constant pressure is defined as

$$\begin{aligned} C_{\mathrm{P}}=\left. \dfrac{\partial H}{dT}\right|_{P}=\left. \dfrac{ \delta Q}{dT}\right|_{P}. \end{aligned}$$

(6.32)

The mass transfer also introduces energy convection into or out of the chamber. The rate of heat convected into the chamber is given as

$$\begin{aligned} \dot{Q}_{\mathrm{i}}=\dot{m}_{\mathrm{i}}C_{\mathrm{P}}T_{0}. \end{aligned}$$

(6.33)

Note that the specific heat at constant pressure, $C_{\mathrm{P}}$, used in the above expression changes depending upon the air–fuel ratio. Likewise, the rate of heat taken out with exhaust is

$$\begin{aligned} \dot{Q}_{\mathrm{o}}=\dot{m}_{\mathrm{o}}C_{\mathrm{P}}T. \end{aligned}$$

(6.34)

Further noting that $\delta Q=dU+PdV$,

$$\begin{aligned} dU=\delta Q-PdV\mathrm{or}\dot{E}=\dot{Q}-P\dot{V}. \end{aligned}$$

(6.35)

The total energy balance equation is then given as

$$\begin{aligned} \dot{E}=\dot{Q}_{\mathrm{c}}+\dot{Q}_{\mathrm{i}}-\dot{Q}_{\mathrm{o}}-P\dot{V} -\lambda _{\mathrm{gb}}\left( T-T_{\mathrm{b}}\right) , \end{aligned}$$

(6.36)

where $\dot{Q}_{\mathrm{c}}$ is the heat released due to combustion, $\lambda _{\mathrm{gb}}$ is the overall heat transfer coefficient between the gas and body (cylinder $+$ piston), and $\lambda _{\mathrm{gb}}\left( T-T_{\mathrm{b} }\right)$ is the heat transferred to the body.

The metallic body also exchanges heat with the environment. The heat balance equation for the metallic body is given as

$$\begin{aligned}&m_{b}C_{\mathrm{p}_{\mathrm{b}}}T_{\mathrm{b}}\left( t\right) =m_{b}C_{\mathrm{p}_{ \mathrm{b}}}T_{\mathrm{0}}+\mathop {\displaystyle \int }\limits _{0}^{t}\left( \lambda _{\mathrm{gb}}\left( T\left( \tau \right) -T_{\mathrm{b}}\left( \tau \right) \right) -\lambda _{ \mathrm{ba}}\left( T_{\mathrm{b}}\left( \tau \right) -T_{\mathrm{0}}\right) \right) d\tau , \\ \nonumber&\mathrm{or\ \ }T_{\mathrm{b}}\left( t\right) =T_{\mathrm{0}}+\dfrac{1}{m_{b}C_{ \mathrm{p}_{\mathrm{b}}}}\mathop {\displaystyle \int }\limits _{0}^{t}\left( \lambda _{\mathrm{gb}}\left( T\left( \tau \right) -T_{\mathrm{b}}\left( \tau \right) \right) -\lambda _{ \mathrm{ba}}\left( T_{\mathrm{b}}\left( \tau \right) -T_{\mathrm{0}}\right) \right) d\tau , \end{aligned}$$

(6.37)

where $m_{b}$ is the mass of the body, $C_{\mathrm{p}_{\mathrm{b}}}$ is the specific heat capacity of the body and $\lambda _{\mathrm{ba}}$ is the overall heat transfer coefficient between the body and the environment.

Fig. 6.13

Bond graph model of heat and mass transfer in the engine

The bond graph model of the heat and mass transfer part of engine model is shown in Fig. 6.13. The two modulated R-elements (MR-elements) model the inlet and outlet flows (as per Eqs. 6.28 and 6.29) and the $0_{H}$ junction models the material balance. The heat balance, given in Eq. 6.36, is modeled at 0$_{T}$ junction. The model of heat transfer to body and then to environment is marked in the figure wherein 0$_{Tb}$ junction models the heat balance in the metallic body, the C-element represents the heat capacity of the metallic body, and resistances R$_{gb}$ and R$_{ba}$ model the heat transfers between gas–body and body–atmosphere, respectively.

1.8.4 Combustion

Combustion of fuel takes place through ignition. In gasoline or Otto cycle engines, the air–fuel mixture is ignited through a spark-plug. In diesel cycle or compression ignition engine, the gas pressure and temperature are tremendously high at the top-dead center after the compression stroke and spontaneous ignition takes place. Combustion causes sudden increase in pressure which drives the piston towards the bottom dead center. This phase is known as the power stroke.

The combustion of fuel releases energy into the gas. The amount of energy released depends upon many factors, e.g., type of the fuel (its calorific value), pressure and temperature of the air fuel mixture, mixture quality, spatial uniformity in distribution, engine speed, and so on. The heat of combustion is the energy released as heat when a hydrocarbon is completely combusted with oxygen under standard conditions. Usually, complete combustion never takes place. Some part of the hydrocarbon fuel is fully combusted to form CO$_{2}$, whereas some part is partially combusted to CO and some fuel remains unburnt. Other parasitic reactions also happen during this time. The combustion phenomenon occurs so fast that it is difficult to model it with other dynamics. The time constant difference is so severe that one needs a multi-scale simulation approach. However, an approximate model can be developed by considering experimental values of combustion efficiency. We assume that an efficiency map is available and decides the combustion efficiency $\eta$ based on the state of the system. Note that combustion efficiency is different from engine efficiency.

We assume that the air–fuel mixture is dry, but some amount of water forms as a by-product of combustion process. Then, the lower heating value (LHV) of the fuel is considered as the heat of combustion. For gasoline, it is typically 44.40 MJ/kg. If $q_{ \mathrm{hc}}$ is the heat of combustion or the calorific value, the heat released during the combustion process is

$$\begin{aligned} Q_{\mathrm{c}}=\eta m_{\mathrm{f}}q_{\mathrm{hc}}, \end{aligned}$$

(6.38)

where $\eta$ is determined from the efficiency map [24] and $m_{ \mathrm{f}}=\int \dot{m}_{\mathrm{f}}dt=\int \alpha \dot{m}_{\mathrm{i}}dt$ is the mass of the fuel contained in the control volume. A detailed model of fuel injection system can be consulted in [1].

Although the combustion process is extremely fast, we still need to introduce a rate of combustion so that we can model it using a bond graph. The rate of combustion is determined by chemical kinetics, i.e., the spontaneity of reactions. The difference between the Gibb’s free energy of the reactants and the products decides the reaction progression rate. There are many substeps involved in a chemical reaction and usually a few of those substeps are the rate determining factors [75]. This issue will be addressed at the end of this chapter. In any case, whatever way the combustion rate is modeled, it makes little change to the overall dynamics of the vehicle. We assume that the reaction rate is constant and it takes place during an extremely small time interval $\Delta t$ (close to an impulse function) so that

$$\begin{aligned} \dot{Q}_{\mathrm{c}}=\left\{ \begin{array}{l} \dfrac{\eta m_{\mathrm{f}}q_{\mathrm{hc}}}{\Delta t}\ \mathrm{for}\ \theta _{\mathrm{t}}<\theta <\theta _{\mathrm{t}}+\Delta t\dot{\theta }, \\ 0\mathrm{otherwise}, \end{array} \right. \end{aligned}$$

(6.39)

where $\theta _{\mathrm{t}}$ is the crank angular position when the piston has just passed the top dead center.

Fig. 6.14

Bond graph model of heat and mass transfer, and combustion phenomenon in the engine

Fig. 6.15

Integrated bond graph model of the single cylinder internal combustion engine

The bond graph model of combustion phenomenon along with the heat and mass transfer is given in Fig. 6.14. The part used to model combustion is given in a block diagram form. The accelerator pedal pressure modifies the $\alpha$ parameter value (fuel–air ratio) and the total fuel inlet during suction is obtained by integrating ($\int$ block) the fuel mass flow rate ($\dot{m}_{\mathrm{f}}$). This information is then used to model the heat input rate due to combustion (MSF:$\dot{Q}_{\mathrm{c}}$) as per Eq. 6.39. Note that the fuel efficiency map is also used in determination of $\dot{Q}_{\mathrm{c}}$. It is assumed that any unburnt fuel is discarded completely with the exhaust. This is why, the state associated with the $\int$ block is initialized to zero at the start of the simulation and every time after the exhaust is completed.

1.8.5 Integrated Model

The complete model of the single cylinder engine is given in Fig. 6.15. It is possible to change the valve timings and convert this model into a four-stroke engine. The engine output is torque applied onto the crank shaft.

The crank shaft is connected to the flywheel and the transmission shaft through a clutch. The torque output from engine blocks can be applied on the crank shaft. A V8 engine configuration has been developed later in this section.

1.9 Gearbox

A typical transmission gearbox for the rear wheel drive vehicle is shown in Fig. 6.16. The input shaft, output shaft, and lay shaft rotate with angular speeds $\omega _{ {1}}$, $\omega _{\mathrm{o}}$ and $\omega _{\mathrm{2-3}}$, respectively. The housing block is mounted on the vehicle body with suspensions and it can roll with angular velocity of $\omega _{\mathrm{b}}$ (although this small rotation can be neglected, it is still included in the model). The roll stiffness and roll damping to the block are $K_{\mathrm{b}}$ and $R_{\mathrm{b}}$, respectively.

Fig. 6.16

A cut-out of a single lay shaft transmission gearbox

The bond graph model of gearbox is shown in Fig. 6.17. It is adopted from the model developed in [28]. The engine torque from the crank shaft acts on the gear inertia $J_{1}$ and on the housing block as a reaction torque. The frictional resistances at the input and output shafts are $R_{\mathrm{i}}$ and $R_{\mathrm{o}}$, respectively. The total reaction torque acting on the housing is summation of reaction torque due to two pairs of gears, reaction torque of input and output torques, and reaction torque due to relative rotation of lay shaft with respect to block housing. All the reaction torques are summed up at the 1$_{\omega _{b}}$ junction (which has multiple copies so as to maintain the clarity of the figure, but can be merged to a single junction) where the rotary inertia of the housing block is modeled.

The roll of the gear box housing and vehicle body roll are constrained by the gear box housing suspension. The linear stiffness and damping of this suspension can be expressed as an effective torsional stiffness and damping, which is modeled by C: $K_{\mathrm{be}}$ and R: $R_{\mathrm{be}}$ elements at the 1-junction representing the relative roll velocity between the vehicle body and the gear box housing.

The relative angular speed of gear1 ($\omega _{\mathrm{1}}$) with respect to housing ($\omega _{\mathrm{b}}$) is transformed to obtain the angular speed of the lay shaft $\omega _{\text{2-3}}$ by using the transformer with modulus $-r_{2}/r_{1}$. Similarly, the relative angular speed of the lay shaft with respect to housing ($\omega _{\text{2-3}}-\omega _{\mathrm{b}}$) is transformed to obtain the angular speed of the output shaft $\omega _{\mathrm{o }}$ by using the transformer with modulus $-r_{3}/r_{4}$. The backlash (see Chap. 3) and gear-mesh stiffness and damping between two pair of gears are modeled as torsional spring and damper in series connection (C: $K_{\text{ i-j}}$ and R: $R_{\text{ i-j}}$, where i and j enumerate the gears). Moreover, the element R: $R_{\text{2-3b}}$ models the reaction from lay shaft bearings. The bearing generates torque due to torsional damping depending on the relative angular velocity between the lay shaft and the gear box housing. More detailed models of the bearings (rolling element bearings and journal bearings) are given in the previous chapter.

Fig. 6.17

Bond graph model of transmission gearbox

1.10 Differential

The power transmission to the wheel axles is achieved through different mechanisms such as fluid coupling (discussed later in this chapter), locked differential, planetary gear set with bypass clutches (also detailed later in this chapter), and controlled clutch modulations. The planetary gear set and differential are the most widely used.

The schematic representation of a planetary gear set or epicyclic differential is shown in Fig. 6.18. The angular velocities are denoted by symbol $\omega$, the radii by $r$ and the subscripts s, c, p, and r, respectively, indicate sun, carrier, planet, and ring gears.

Fig. 6.18

Schematic representation of a planetary gear set

The angular velocity of the planet as seen from a fixed frame fixed to the ring gear is the sum of the angular velocity of the carrier and the angular velocity of the planet as seen from a frame fixed to the carrier. The relative angular velocity between the carrier and the sun gear is related to the planet’s angular velocity as follows:

$$\begin{aligned} \dfrac{\omega _{\mathrm{c}}-\omega _{\mathrm{s}}}{\omega _{\mathrm{c}}+\omega _{ \mathrm{p}}}=-\dfrac{r_{\mathrm{p}}}{r_{\mathrm{s}}}. \end{aligned}$$

(6.40)

Likewise,

$$\begin{aligned} \dfrac{\omega _{\mathrm{c}}+\omega _{\mathrm{p}}}{\omega _{\mathrm{c}}-\omega _{ \mathrm{r}}}=\dfrac{r_{\mathrm{r}}}{r_{\mathrm{p}}}. \end{aligned}$$

(6.41)

From these kinematic relations, the bond graph model of the planetary gear set is constructed as shown on the left of Fig. 6.19. The planets are not separately powered and their inertia and resistance to rotation may be neglected. Thus, the 0$_{P}$ junction may be omitted from the model and the reduced model is shown on the right of Fig. 6.19. This model may be arrived at in a straightforward manner by combining Eqs. 6.40 and 6.41 so that the kinematic relation used to create the model becomes

$$\begin{aligned} \dfrac{\omega _{\mathrm{c}}-\omega _{\mathrm{s}}}{\omega _{\mathrm{c}}-\omega _{ \mathrm{r}}}=-\dfrac{r_{\mathrm{r}}}{r_{\mathrm{s}}}\mathrm{or}\dfrac{\omega _{\mathrm{s}}-\omega _{\mathrm{c}}}{\omega _{\mathrm{c}}-\omega _{\mathrm{r}}}= \dfrac{r_{\mathrm{r}}}{r_{\mathrm{s}}}. \end{aligned}$$

(6.42)

Fig. 6.19

Bond graph model of planetary gear set

The negative sign in the TF modulus may be explained as follows: assume that the carrier is locked ($\omega _{\mathrm{c}}=0$). Then the planets will rotate opposite to the sun. The ring gear has internal gear teeth which means it will also rotate in the opposite direction to the sun gear’s rotation.

The bond graph model shown in Fig. 6.19 can be drawn in various forms by using transformer equivalences. For example, if TF-elements are introduced all around the topmost 0-junction, the resulting equivalent model assumes the form shown in Fig. 6.20.

Fig. 6.20

Equivalent bond graph model of planetary gear set

The bevel gear differential or conventional differential is similar to the planetary gear system. The sun and the ring gears are the two bevel gears and the carrier holds the planets. A general schematic representation of the differential with unequal bevel gears and its bond graph model are shown in Fig. 6.21. The bond graph model is derived from the kinematic relation

$$\begin{aligned} \dfrac{\omega _{\mathrm{c}}-\omega _{1}}{\omega _{\mathrm{c}}-\omega _{ 2}}=-\dfrac{r_{2}}{r_{1}}\mathrm{or}\dfrac{\omega _{1}-\omega _{\mathrm{c}}}{\omega _{\mathrm{c}}-\omega _{2}}= \dfrac{r_{2}}{r_{1}}. \end{aligned}$$

(6.43)

Fig. 6.21

Differential and its bond graph model

In most differentials, $r_{\mathrm{1}}=r_{\mathrm{2}}$ which gives $\omega _{ {1}}-\omega _{\mathrm{c}}=\omega _{\mathrm{c}}-\omega _{\mathrm{2}}$ or $\omega _{\mathrm{1}}+\omega _{\mathrm{2}}=2\omega _{\mathrm{c}}$. Thus, the bond graph model of symmetric differential reduces to the form shown in Fig. 6.22.

Fig. 6.22

Reduced bond graph model of a symmetric differential

This bond graph model explains a lot of physics of the differential. It is understood from the model that half of the engine torque is transmitted to each wheel. Furthermore, if any of the wheels are free to rotate (e.g., slips on an icy road) then the reaction torque from that wheel becomes zero, i.e., the torque on the other wheel also becomes zero. Consequently, the other wheel does not spin and the slipping wheel spins at the same speed as that of the input shaft.

From the preceding discussion, we find that if one wheel of a vehicle does not develop any frictional tire force then the other wheel which may be on a surface with very good grip cannot develop any driving force. As a result, the vehicle cannot move. To remedy this problem, some differentials have a clutching mechanism. If any wheel slips beyond a threshold value, the corresponding bevel gear in the differential is clutched to the carrier. Thus, the differential is controlled by the wheel slip. Such type of differential is called a limited-slip differential.

1.11 Transmission Line

The bond graph model of a complete vehicle with V8 engine is shown in Fig. 6.23. Engines E$_{1}$ to E$_{8}$ are connected to the crank shaft. The crank shaft drives the cam which is assumed to consume little power and hence is modeled in the signal domain.

Fig. 6.23

Integrated model of a vehicle with engine and transmission line

The state vectors associated with the eight engines are to be initialized appropriately. If the piston of the first engine is assumed to be at top dead center at the start-up then the position for successive engines has to be changed accordingly so as to correspond with $45^{\circ }$ crank rotation. The initial volumes are calculated from piston positions and then the initial gas mass and energy are calculated.

All vehicles require some kind of start-up device to rotate the engine. Usually, there is a start-up electric motor (called self-motor) which draws power from the electric battery. We have considered that the self-motor initially supplies a fixed angular velocity ($\omega _{\mathrm{st}}$) for a few cycles of engine rotation. The detailed model of self-motor can be developed and appended to the model given in Fig. 6.23. A flywheel is used on the crank shaft to maintain constant output speed. Without the flywheel, the engine would fail to rotate when the self-motor is turned off. The flywheel is an accumulator of energy which allows the engine to rotate through the bottom and top dead centers without getting locked there. The flywheel rotary inertia is modeled in Fig. 6.23 by I: $J_{ \mathrm{fw}}$ element.

The engine is coupled to the gear box through a clutch. The gear box and the differential are suspended from the vehicle body. The clutch may be of different types and it may be absent in some designs (see the fluid-coupling and power-split device in this chapter). In mechanical friction clutch, usually dry-friction and stick-slip friction models [33] are used. The clutch torsional stiffness and damping are modeled by C: $K_{ \mathrm{c}}$ and R: $R_{\mathrm{c}}$ elements in the bond graph model. The values of parameters $K_{\mathrm{c}}$ and $R_{\mathrm{c}}$ depend upon the pressure on the clutch pedal (unclutched, partially clutched, or fully clutched state). In a detailed friction model, the clutch force is modeled according to the normal pressure and the effective coefficient of friction which is a function of the slip ratio (the ratio of output to input speed). Details of such friction models are given earlier in this chapter during discussions on tire forces.

In Fig. 6.23, C: $K_{\mathrm{t}}$ and R: $R_{\mathrm{t}}$ elements model the torsional stiffness and damping offered by the long transmission shaft between the gear box (usually located at the front of the vehicle) and the differential. The universal coupling and other minor components have been neglected in this model.

1.12 Engine Dynamics Simulation

The engine is simulated at 30 % throttle position when the engine is disengaged from the driveline. The fuel injected is high at the engine start-up and then it is regulated as per the throttle position. The self-motor is operated for first three engine cycles at high fuel load. The crank shaft is connected to a large flywheel and clutches separate the engine from any other load. The engine parameters [7] are given in Table 6.1.

Table 6.1 Parameter values used in simulation of the V-8 engine model

The time evolution of gas pressure in two engines is shown in Fig. 6.24a. The variation in crank shaft speed and torque during the initial start-up stage is shown in Fig. 6.24b. The time gap between the two pressure responses in the first cycle (d$_{1}$) is large. As the engine speed increases, this phase gap reduces. Gradually, a steady engine speed is reached.

Fig. 6.24

Simulated dynamics of engine start-up

The spikes in the crank-shaft speed and torque seen in the initial phase are due to the self-motor action. Once the crank-shaft has started rotating at a sufficient speed, the kinetic energy stored in the flywheel or other components of the transmission line are sufficient to sustain the engine revolution. The self-motor is then switched off.

The steady crank-shaft speed and torque are shown in Fig. 6.25a. Because only an inertia load is there, at steady state the engine torque is nearly zero (except some little torque required for overcoming damping forces). This is a no-load engine simulation.

Fig. 6.25

a Engine steady-state results and b P-V diagram

The pressure–volume (P-V diagram) curve for one of the engines is shown in Fig. 6.25b where the path between a to b is the expansion stroke, path b to c is the suction and discharge phase, path c to d is the compression stroke, and path d to a is combustion phase. The inset in Fig. 6.25b shows the details during suction and exhaust phases taking place around 1bar cylinder pressure. The exhaust takes place in the path between point B and point A in the inset. During the path between points A and B in the inset, the pressure difference is not conducive to either exhaust or suction. From point B to point C in the inset, suction takes place.

The steady-state P-V diagram for all engines is the same. Note that the steady-state maximum pressure is only about 40 bars because the engine is running unloaded. Moreover, the thermodynamic efficiency is less in this condition. The maximum cylinder pressure increases to around 80–100 bars under full throttle (first gear) and load.

If a 4-stroke engine is simulated then the PV-diagram becomes different. The expansion stroke is from point a (TDC or top dead center) to point c (BDC or bottom dead center). Then in the exhaust stroke, the piston moves from BDC to TDC with little change in pressure (marginally above atmospheric). The suction stroke takes the cylinder from TDC to BDC where the pressure also remains more or less constant (a little above atmospheric) and it is then followed up by compression stroke (BDC to TDC) and combustion. Thus the four-stroke engine’s PV-diagram has a smooth path between points a to c followed by two approximately constant pressure lines from BDC to TDC and back to BDC and finally the path from points c to a via point d.

1.13 Clutch

A clutch is used to connect the driving shaft to a driven shaft, so that the driven shaft may be started or stopped at will, without stopping the driving shaft. Thus, a clutch provides an interruptible connection between two rotating shafts. A clutch allows a high inertia load to be started with a small power. The clutch allows disengagement of the drive when brakes are applied on the driven shaft and thus continue the engine to run at no load. Clutches are also used during gear change so that sudden transient loads on the engine are avoided.

Clutches are of various kinds: friction clutch, dissipative viscous coupling, torque converter, etc. The detailed modeling of torque converter will be discussed later in this chapter. For the time being, a clutch can be modeled as a device which produces slip-dependent torque. The basic bond graph model of a clutch is a modulate R-element where the modulating signal indicates the pressure applied on the clutch. In a dry friction clutch, the friction pad is made of cotton and asbestos fibers woven or molded together and impregnated with resins or other binding agents. In many friction discs, copper wires are woven or pressed into the facings to give them added strength. Some friction discs use ceramic-metallic facings. Grooves are cut on the disk to assist cooling and also to avoid vacuum creation at high speeds such that disengagement is not hampered.

The schematic drawing of a friction clutch (a single plate dry clutch) is shown in Fig. 6.26 where the clutch plate is interposed between the flywheel surface of the engine and pressure plate. The engine flywheel, a friction disc called the clutch plate and a pressure plate are the basic components of a clutch. When the pressure plate is attached to the flywheel it rotates at the same speed as the crankshaft. When the driver pushes down the clutch pedal the pressure plate moves away from the friction disk and power transmitted through the clutch is interrupted. When the driver releases the clutch pedal, power is transmitted through the clutch. Springs in the clutch push the pressure plate against the friction disk. This pushing force determines the maximum frictional force between the friction disk and the flywheel, and determines the slip between the two. In separate arrangements, more number of mating surfaces are used for transmitting the torque.

Fig. 6.26

Schema of a friction clutch

Two basic types of clutch are the coil-spring clutch and the diaphragm-spring clutch. The coil spring clutch, which has a series of coil springs set in a circle, uses coil springs as pressure springs. At high rotational speeds, centrifugal forces on the coil springs and the lever of the release mechanism cause problems in the operation of this kind of clutch. Diaphragm type springs are used to avoid such problems. In diaphragm type design, the centre plate, on which the friction facings are mounted, consists of a series of cushion springs. These cushion springs are radially crimped (wavy) so that the crimping is progressively squeezed flat as the clamping force is applied to the facings. This arrangement enables gradual transfer of the force. When the clamping force is released, the plate springs back to its original crimped state. The power is transmitted from the plate to the hub through coil springs interposed between them. These springs are carried within rectangular holes or slots in the hub and plate. These heavy coil springs set in a circle around the hub smooth out the torsional vibrations and reduce transient loading of the plate and the engine.

Two basic formulations are followed to model friction clutches. Uniform pressure assumption is made for a new clutch whereas uniform wear is assumed for a worn out clutch.

Fig. 6.27

Elemental area on friction plate

If the friction disk is an annulus with external radius $R_{ \mathrm{o}}$ and the internal radius is $R_{\mathrm{i}}$ (Fig. 6.27) then the uniform pressure distribution yields $p_{\mathrm{0}}=N/\left( \pi \left( R_{o}^{2}-R_{i}^{2}\right) \right)$ where $N$ is the force applied on the clutch. The maximum transmissible torque is then given as

$$\begin{aligned} T_{\max }=\mathop {\displaystyle \int }\limits _{R_{i}}^{R_{o}}\mathop {\displaystyle \int }\limits _{0}^{2\pi }\mu _{s}p_{0}r^{2}\mathrm{d}\theta \mathrm{d}r=\dfrac{2\mu _{s}}{3}\left( \dfrac{ R_{o}^{3}-R_{i}^{3}}{R_{o}^{2}-R_{i}^{2}}\right) N. \end{aligned}$$

(6.44)

where $\mu _{s}$ is the coefficient of static friction. For dynamic modeling, a stick slip formulation is used or Stribeck effect is separately included. A better way is to use empirical models (like tire force calculation with Pacejka or Burckhardt formulations given in Eqs. 6.3 and 6.4, respectively). Thus, the clutch model is given by slip ratio instead of absolute value of the slip. As a consequence, the bond graph model is a two-port modulated R-field where the modulating signal is the applied force. The slip at any point at distance r away from the center of the annulus and angle $\theta$ from a reference radial line is given as $\sigma =\left( \omega _\mathrm{i}r-\omega _\mathrm{o}r\right) /\omega _\mathrm{i}r=\left( \omega _\mathrm{i}-\omega _\mathrm{o}\right) /\omega _\mathrm{i}$, where $\omega _\mathrm{i}$ is the input shaft speed and $\omega _\mathrm{o}$ is the output shaft speed. Thus, the slip ratio is constant at all points on the surface, which means the instantaneous coefficient of friction at a given slip ratio is also constant. Thus, the instantaneous torque transmitted is in given as

$$\begin{aligned} T\left( \sigma \right) =\dfrac{2\mu \left( \sigma \right) }{3}\left( \dfrac{ R_{o}^{3}-R_{i}^{3}}{R_{o}^{2}-R_{i}^{2}}\right) N. \end{aligned}$$

(6.45)

In uniform wear assumption, the wear rate is given according some established theories for the wear in a mechanical system. It is assumed that the wear rate $R_{w}$ is proportional to the $pV$ factor where $p$ refers the contact pressure and $V$ the sliding velocity. Then

$$\begin{aligned} R_{w}=kpV \end{aligned}$$

(6.46)

where $k$ is a constant of proportionality. If the relative angular velocity between the drive and driven shaft $\omega =\omega _\mathrm{i}-\omega _\mathrm{o}$ then the relative linear velocity at any point on the face of the clutch at distance $r$ from the centre of the annulus is $V=\omega /r$. Thus, combining these equations and assuming a constant angular velocity $\omega$

$$\begin{aligned} R_{w}=kp\omega r\Rightarrow pr=R_{w}/\left( k\omega \right) =K \end{aligned}$$

(6.47)

where $K$ is another constant. The largest pressure $P_{\max }$ occurs at the smallest radius $R_{i}$ and thus $K=p_{\max }R_{i}$. The pressure at any point at a distance $r$ from the centre of the annulus is, thus,

$$\begin{aligned} p=p_{\max } \dfrac{R_{i}}{r}. \end{aligned}$$

(6.48)

The axial force N is obtained as

$$\begin{aligned} N&=\mathop {\displaystyle \int }\limits _{R_{i}}^{R_{o}}\mathop {\displaystyle \int }\limits _{0}^{2\pi }p_{\max }\dfrac{ R_{i}}{r}r\mathrm{d}\theta \mathrm{d}r=2\pi p_{\max }R_{i}\left( R_{o}-R_{i}\right) \\ \nonumber&\Rightarrow p_{\max }=\dfrac{N}{2\pi R_{i}\left( R_{o}-R_{i}\right) }. \end{aligned}$$

(6.49)

The maximum transmissible torque is calculated as

$$\begin{aligned} T_{\max }&=\mathop {\displaystyle \int }\limits _{R_{i}}^{R_{o}}\mathop {\displaystyle \int }\limits _{0}^{2\pi }\mu _{s}p_{\max }\dfrac{R_{i}}{r}r^{2}\mathrm{d}\theta \mathrm{d}r=\mu _{s}\pi p_{\max }R_{i}\left( R_{o}^{2}-R_{i}^{2}\right) \\ \nonumber&\Rightarrow T_{\max }=\mu _{s}N\left( \dfrac{R_{i}+R_{o}}{2}\right) =\mu _{s}NR_{av} \end{aligned}$$

(6.50)

where $R_{av}$ is the average radius of the friction pad annulus.

Because the slip ratio is constant, the formulation based on slip ratio yields instantaneous torque transmission as

$$\begin{aligned} T\left( \sigma \right) =\mu _{s}\left( \sigma \right) NR_{av}. \end{aligned}$$

(6.51)

1.13.1 Clutch Pressure Control

The pressure plate (See Fig. 6.26) is moved in appropriate direction to control the normal reaction at the friction planes. This actuation is usually performed through electro-hydraulic valve actuator, which is also called an amplifier valve. The schema of an amplifier valve is shown in Fig. 6.28. In this figure, the clutch model has been simplified to include only the essential dynamical parts [28].

The solenoid is controlled through pulse width modulated (PWM) current signal. In some recent designs of amplifier valves used in direct electronic shift control, which use more powerful solenoid to provide more flow throughput, the spool is not required. Such designs are used in active suspensions and antilock brake systems (ABS), which are described in details in later sections of this chapter.

Fig. 6.28

Cut-out of an amplifier valve

The model of the solenoid has been already developed in Chap. 3. Suitable control of the duty cycle of the PWM solenoid valve sets the plunger displacement which causes variations in the solenoid cavity pressure. When the spool valve is moved to extreme right position in Fig. 6.28, the resistance to flow between the line (port P or pump) and the clutch (port C) becomes small and thus maximum pressure is applied on the clutch. When the spool valve moves to extreme left position in Fig. 6.28, the flow path between the line and the clutch is closed, but the flow path between the line (port P) and the vent (V) is fully open. This causes minimum pressure on the clutch and disengages the clutch. In intermediate positions, the leakage resistances decide the amount of pressure applied on the clutch. The position of the spool valve is controlled by the plunger.

The bond graph model of this electro-hydraulic actuator [28] is given in Fig. 6.29. The plunger/solenoid displacement and line pressure are the inputs to this model. The solenoid displacement is obtained from the model of the solenoid’s electro-magnetic domain (See Chap. 3). The plunger position controls the passage resistances (modulated R-elements) between the pump and the cavity and the cavity and the sump. The cavity fluid compliance is modeled by element $C_{s}$. The pump (line) and sump pressures are denoted by P$_{l}$ and P$_{0}$ , respectively. The transformers with moduli $A$ and 1/$A$, where $A$ is the exposed cross-sectional area of the spool where the pressure acts, transform the pressure to force on the spool. The spool inertia, compliance and frictional resistance are modeled for the spool dynamics. The pressure of the fluid in the spool chamber (between line, clutch and vent) is represented at junction 0$_{c}$, where C$_{c}$ models the fluid compliance. Two modulated R-elements control the passage resistances between the spool chamber and the clutch cavity and between the spool chamber and the vent. The modulating signal is the position of the spool. The flow through the hydraulic line (pipe) between the spool chamber and the clutch cavity is modeled by accounting for the fluid inertia and damping (skin-friction and viscous losses). The fluid compliance in clutch cavity is modeled by C$_{c}$ element and R$_{l}$ represents leakage resistance. The pressure in the clutch cavity applies force on the clutch plate. In the bond graph model, $A_{\mathrm{p}}$ is the effective frontal area of the pressure plate, and the I-, C-, and R-elements used to model the clutch plate represent the clutch or piston inertia, effective structural compliance (including the return spring) and the structural damping, respectively.

Fig. 6.29

Bond graph model of clutch pressure control

An actively controlled wet clutch allows easier control of torque transfer. It is a key part of active limited slip differential which significantly improves the vehicle dynamics and traction control. A detailed bond graph model of active limited slip differential can be consulted in [30].

1.13.2 Automatic Clutch During Gear Shift Process

Manual clutching is required during gear shifting. An automatic transmission system uses a torque converter whose details are given later in this chapter. A schematic of a four-speed automatic transmission system is shown in Fig. 6.30a (torque converter is excluded). It consists of reverse, low and high range clutches. A single shift actuator called a dog actuator is used in this design [28]. This type of design is simple, easy to maintain and does not require synchronizers. The turbine of the torque converter effectively acts as the synchronizer. The dog actuator is positioned appropriately (on the splined shaft) to either engage with the second or the third gear or with none of them. The first and fourth gear operations are performed by engaging the respective clutches. The output is then connected to the differential.

Fig. 6.30

a Clutch actuator in automatic transmission and b torque and speed drop transients during shifting process

In this type of design, the torque drop can be significant during a slow shift process. That is why electronic (microcomputer controlled) shifting mechanisms are used. These shifting mechanisms ensure quick gear shift with minimal torque-drop (See Fig. 6.30b). Moreover, it can be tuned to adapt the shift execution and shift scheduling according to different driving conditions. Faster shift is recommended to improve fuel efficiency whereas slower shift gives better drivability.

The bond graph model of the shift system in an automatic transmission system shown should include the inertias at the input and output of the transmission system and should also include the hydraulic control systems required to engage the clutches and the dog actuator. The effective inertia between the input and output of the transmission system can be represented as a two-port I-field. In this case, there is no inertia cross-coupling and hence the off-diagonal terms in the mass matrix are zero (in effect, we have two independent single port I-elements). In the bond graph model given in Fig. 6.31, the transformers model various gear ratios (indicated by first subscripts u and d for up and down sides, respectively, and second subscripts h, l and 2/3 for low, high and second/third gears, respectively). The gear mesh stiffness and damping are modeled for second and third gears (dog actuator part) so that the model remains in integral causality. Note that transformer moduli $\mu _{u2/3}$ and $\mu _{d2/3}$ have to be switched suitably depending upon the dog actuator position, e.g., to zero when the dog actuator does not engage with either second or third gears. The modulated R-elements model the clutches where the pressures applied on the clutches are the modulating variables. The clutches thus allow different amounts of slip between the up and down sides of the automatic transmission device depending upon the pressure applied by the amplifier valves. The bond graph model given in Fig. 6.31 is applicable to both layshaft automatics and planetary automatic transmission (see later sections for details) with clutches and bands [12].

Fig. 6.31

Bond graph of clutch actuator in automatic transmission

Fig. 6.32

Bond graph model of differential

1.14 Integrated Four Wheel Vehicle Model

The post-clutch section of a rear-wheel driven vehicle is modelled in Fig. 6.32 as a source of effort. The input torque is applied at a 0-junction representing the differential, which implies same torque but different velocities in the two traction wheels. The main gear box and differential gearbox (two MTFs) are not modeled here.

Note that the modular model developed here keeps the option open to include detailed engine and transmission system models.

1.15 Simulation Results for Four Wheel Model

The normal loads or load transfer on the wheels vary due to pitch, yaw and roll motions. At high speeds, aerodynamic forces can as well affect the load transfer. Therefore, the load on each wheel, which is the deciding factor of maximum braking and driving torques, is influenced by various factors; broadly the operational conditions (speed, steering angle, $\ldots$) and the environmental conditions (wind speed, road condition, $\ldots$). Initially, to get an idea about the handling dynamics of the vehicle, vehicle centre of mass trajectory in inertial frame and variation of normal loads on the wheels are studied. The parameter values used in the simulation of full vehicle model are taken from [6]. The developed vehicle model is further validated by comparing the results with experimental manoeuvre data available in literature [8].

1.15.1 Steering Response of the Vehicle

It is fundamental requirement that a vehicle should show under-steer characteristics above a certain value of linear speed. This would help the driver in safe maneuvering of the vehicle at high speeds. On the contrary, if a vehicle starts over-steering then the driver needs to quickly make clockwise and anticlockwise steering wheel rotations, which is both uncomfortable and unstable. Here, the vehicle operating speed is chosen as 50 km/h (13.88 m/s) and we desire the vehicle to under-steer at this speed.

For simulating the steering response, the vehicle is steered with a fixed steering angle of 0.1 rad after an initial straight run for 12 s by which time the vehicle has reached a steady speed of 13.88 m/s. The variations in wheel speeds during maneuvering are shown in Fig. 6.33a. The result shows under-steer characteristics (Fig. 6.33b) with steady mean speed of 13.67 m/s and 0.446 m/s difference in speeds between inner and outer wheels. At steady turning radius of $r=42.9$ m, kinematic analysis with half track width $c = 0.7$ m gives difference in wheel speeds as $2c \sqrt{\dot{x}_{\mathrm{v}}^{2}+\dot{y}_{\mathrm{v}}^{2}}/r=2\times 0.7\times 13.67/42.9=0.446$ m/s which matches the simulated results.

1.15.2 Slalom Manoeuvre

The test data of [19] are used validate the developed four wheel model. In [19], slalom manoeuvre test was carried out on the vehicle at fixed vehicle speed of 50 km/h (13.88 m/s) and the steering wheel angle and the yaw rate were measured. The steering wheel angle was maintained between ${\pm }100^{\circ }$ and the gear ratio between the steering wheel and axle rotation was 25:1. With this small change in axle orientations, the under-steered vehicle’s linear speed remains almost constant at 50 km/h.

Fig. 6.33

a Speeds for left and right front wheels; and b Vehicle centre of mass trajectory in inertial frame

The experimental steering angle values ($\delta _{ \mathrm{stw}}$ in Fig. 6.34) at fixed time intervals were obtained from the graphs in [19] and were used as input ( $\dot{\theta }_{\mathrm{ref}}=d\delta _{\mathrm{stw}}/dt$) in the developed four wheel bond graph model. The simulated yaw rates are compared with the experimentally obtained yaw rates of [19] in Fig. 6.34. Moreover, the results obtained in [19] from a reduced order vehicle model are also plotted in Fig. 6.34 for comparison. It is found that the two models accurately predict the actual handling behavior of the vehicle.

Fig. 6.34

Comparison of vehicle’s yaw rate during slalom maneuver at vehicle speed of 50 km/h with the test and simulation results of Drozdz and Pacejka

2 Suspension Systems

Vehicle suspension systems are designed to provide good riding comfort and manoeuvrability which means reduction of the force transmitted to the driver/vehicle while maintaining good tire contact with the ground. Better riding comfort requires a suspension system with a low natural frequency. On the other hand, to have good handling, a suspension system should have high natural frequency. In order to have a comfortable ride, the suspension system should have a low natural frequency which would filter out the high frequency inputs on rough roads. But to have good handling, the vehicle should have a suspension system with a high natural frequency, so as to filter out the relatively low-frequency cornering motions. Weight or load transfer under braking or acceleration, which causes pitching movements, known as dive or squat depending on their direction, is another point to consider during a suspension design. The load transfer between the inner and outer wheels while negotiating a curve also influences suspension performance. These contrasting requirements have been addressed from many angles to develop various advanced passive, active and semi-active suspension systems. Various solutions to the conflicting requirements have been proposed. The development of vehicles with a lower centre of gravity, springs with rising stiffness, added springs to increase stiffness in roll without increasing it too much in bump, all have led to much improved ride and handling of vehicles. But as the demand for improved fuel economy, greater payload, better on- and off-road ability, higher speed and for greater comfort, continue to increase in a competitive automobile market, the designer needs to continually optimize the suspension performance and find a solution to the ride/handling constraint according to the prevailing market demand.

Vehicle suspensions are designed not only to reduce the shock fed back to the rider and machine, but also to keep the wheels in better contact with the ground. A pneumatic tire acts like an air spring and little damping is offered due to the deformation of the rubber. Therefore, entire effective damping to road inputs is provided by the suspension system between the axle and the vehicle body. Consider a vehicle suspension without dampers. When the wheel goes over a bump, the suspension spring is compressed and then it causes the vehicle body to go up. The vehicle body continues to go up and pulls the wheels up with it, even when the wheel has crossed over the bump. Then the vehicle body falls under it own weight from the peak height and the up-down motion continues. Thus, tire-road contact is lost in some periods which means loss of traction and steerability. The suspension damper is used to absorb the potential energy stored in the suspension spring when the vehicle encounters a bump so as to avoid excessive vertical motion of the vehicle body and also loss of tire contact with the ground. However, dampers and springs must be correctly matched to each other. If the damping is too little then the wheel continues bouncing for long durations. On the other hand, too much damping reduces the ride comfort because the damper acts as a rigid strut when excitation frequencies are large and transmits the entire road disturbance to the vehicle body.

Suspension systems may be broadly divided into two categories: passive systems composed of springs and dampers where the stiffness and damping may be controlled with little use of external power and active systems where suspension forces are completely generated by actuators requiring large amounts of power from an external source. Some suspension systems are mixed in the sense that large amount of power from the external source may be needed for shorter time durations and very little power is required to operate the suspension at other times.

Fig. 6.35

Elements of passive suspension system

Passive suspension is composed of springs, shock absorbers and linkages that connects the wheels (or axles) to the vehicle. There are various types of passive suspensions constructions. The spring may be a coil spring, leaf spring, torsion beam, air/liquid spring, etc. The shock absorber gives damping to the system and usually, the energy is dissipated by forcing hydraulic fluid through orifices. The suspension configuration may be dependent type or independent type. In independent suspensions, motion of the wheels on opposite sides are not coupled. Examples of dependent suspension type are Watt’s linkage and Panhard rod. Examples of independent suspension type are MacPherson strut, double wishbone suspension, multi-link suspension, swinging arm suspension, etc. The general schematic model of a suspension system is shown in Fig. 6.35, which is often called a quarter car model. In this model, the sprung mass refers to one fourth of the vehicle body mass (including passengers and other loads) and unsprung mass refers to the mass of the wheel and the axle. The pneumatic tire usually has a much larger stiffness and low damping coefficient. Therefore, only tire stiffness is considered in the quarter car model.

Designers have been continuously striving to optimize the suspension performance for better handling and rider comfort, and higher payload. There is a considerable body of literature related to the design of vehicle suspension systems in both passive and active modes. Passive systems can be non-adjustable and non-controllable, or adjustable and non-controllable. Active systems provide a range of adjustment and controllability that provide a quality ride over a wide range of road conditions. These systems can be mechanical, hydro-mechanical, hydro/pneumatic-mechanical, and pure hydraulic in various configurations. Active suspension technology allows to change the suspension’s natural frequency and damping in real time depending upon user preferences and thereby it provides a solution to the ride/handling constraint. Active components consume very little power, which is spent in driving control mechanisms for stiffness and damping selection. Usually, auxiliary stiffness and damping chambers are provided in air and liquid springs and the stiffness / damping selection mechanism consists of a set of controlled valves.

2.1 Passive Liquid-Spring Shock Absorber

Liquid shock absorbers are compact, light weight suspension struts which reduce the suspension play and they are generally used in very heavy load applications, such as heavy military vehicles, landing gear suspension in aircrafts and space shuttles. The schematic diagram ² of a liquid spring is shown in Fig. 6.36. The liquid flow through orifices produces damping, whereas; the cushioning effect comes from the fluid’s compressibility.

Fig. 6.36

Liquid spring schematic diagram

Silicone based fluids, which have smaller bulk modulus, are non-corrosive, and are considered stable, are considered suitable for liquid springs.

2.1.1 Mathematical Model

It is usually sufficient to assume that the thermodynamics plays a very minor role in liquid spring dynamics. Therefore, one may assume isothermal operation. The internal damping in compressible liquids is dependent on the strain-rate. This leads to the direction dependent instantaneous anisotropic pressure. The resultant force on the piston is a superimposition of force due to isotropic fluid pressure ($P$) and rate dependent pressure ($P_{d}$). The isotropic pressure leads to elastic force and the anisotropic pressure leads to damping force on a surface.

Change in isotropic pressure developed due to change of volume is given by

$$\begin{aligned} \Delta P=-\beta \left( \frac{\Delta V}{V_{0}}\right) =-\beta \frac{\left( \frac{m}{\rho _{0}} -V\right) }{\left( \frac{m}{\rho _{0}}\right) }=-\beta \left( 1-\frac{\rho _{0}V}{m}\right) =-\beta \left( 1-\frac{\rho _{0}}{\rho }\right) , \end{aligned}$$

(6.52)

where $\Delta V$ is the reduction in volume with respect to free space volume $V_{0}$, $\rho$ is the instantaneous fluid density, $\rho _{0}$ is the free space fluid density, $m$ is the mass contained within the control volume, and $\beta$ is the fluid’s Bulk modulus. Taking free space pressure as reference, the gage isotropic pressure (compression is taken as positive pressure) $P=-\Delta P$.

From classical fluid mechanics, we know that

$$\begin{aligned} \left( 2\mu +3B\right) \nabla .q=\sigma _{xx}+\sigma _{yy}+\sigma _{zz}+3P, \\ \nonumber \nabla .q=\dot{\epsilon }_{xx}+\dot{\epsilon }_{yy}+\dot{\epsilon } _{zz},\mathrm{and}\,B=-\frac{2}{3}(\mu -\mu _{1}), \end{aligned}$$

(6.53)

where $q$ is the velocity vector of stream lines, $\mu$ is the coefficient of viscosity, $\mu _{1}$ is referred to as the second viscosity coefficient or elongational viscosity, $B$ is a dependent parameter, $\sigma _{xx},\sigma _{yy},\sigma _{zz}$ are the normal stresses in Cartesian coordinate, $\epsilon _{xx},\epsilon _{yy},\epsilon _{zz}$ are the linear strains in Cartesian coordinate, and superposed ‘.’ denotes time derivative, i.e. $\dot{\epsilon }_{xx},\dot{\epsilon }_{yy},\dot{\epsilon }_{zz}$ are rate of linear strains. For isotropic case, i.e. incompressible fluids,

$$\begin{aligned} \sigma _{xx}+\sigma _{yy}+\sigma _{zz}=-3P. \end{aligned}$$

(6.54)

The anisotropic pressure or stress on an infinitesimal element is given by

$$\begin{aligned} \sigma _{ij}=\mu \left( \frac{\partial q_{i}}{\partial x_{j}}+\frac{\partial q_{j}}{\partial x_{i}}\right) +\mu _{1}\delta _{ij}\nabla .q, \end{aligned}$$

(6.55)

where $\delta$ is the Kronecker delta. The component of the viscous force ($F_{i}$) per unit volume (total volume $V$) in the direction of the rectangular coordinate $x_{i}$ in a small cubical fluid element is given by

$$\begin{aligned} \frac{F_{i}}{V}=\frac{\partial }{\partial x_{j}}\left[ \mu \left( \frac{ \partial q_{i}}{\partial x_{j}}+\frac{\partial q_{j}}{\partial x_{i}}\right) +\mu _{1}\delta _{ij}\nabla .q\right] . \end{aligned}$$

(6.56)

Note that $\nabla .q=\partial \dot{x}/\partial x=\partial \dot{V}/\partial V$, where x is the position of the fluid element, and $\partial \dot{V} /\partial V$ is the local volumetric strain rate. For unidirectional flow without shear (due to movement of piston) in direction x, Eq. 6.55 is written as

$$\begin{aligned} \sigma _{xx}=\mu _{1}\frac{\partial \dot{x}}{\partial x}. \end{aligned}$$

(6.57)

A linear velocity gradient for the flow stagnation inside the chamber is assumed with boundary conditions $\dot{x}(0)=0$ and $\dot{x}(x_{p})=\dot{x}_{p}$, where the position of the piston with respect to the casing is $x_{p}$ and the velocity of the piston is $\dot{x}_{p}$. Assuming Newtonian fluid, Eq. 6.57 reduces to

$$\begin{aligned} P_{d}=-\sigma _{xx}=-\mu _{1}\left( \frac{\dot{x}_{p}}{x_{p}}\right) =-\mu _{1}\left( \frac{\dot{V}_{p}}{V_{p}}\right) , \end{aligned}$$

(6.58)

where V$_{p}$ is the control volume. Similar integer and non-integer order non-linear relations can be derived for Stoksian fluids. For the liquid-spring, Eq. 6.58 can be represented in a bond graph model as a state dependent damper ($R$-element) with damping coefficient equal to $\mu _{1}/V_{p},$ $p=1,2$. Note that $\mu _{1}=\mu B-\frac{2}{3}\mu ,$where $\mu B$ is the bulk viscosity, and $\mu _{1}$ is a non-linear function of density, $\rho _{p}$ (ratio of the two state variables, i.e. $m_{p}/V_{p},$ $p=1,2$).

2.1.2 Spring Stiffness

When the pressure on both sides of the piston is same, the net resultant force $(F)$ is the difference between the forces acting on the top and bottom surfaces of the piston. Thus,

$$\begin{aligned} F=PA_{p}-P\left( A_{p}-A_{r}\right) =PA_{r}. \end{aligned}$$

(6.59)

For a single liquid spring (see Fig. 6.36), the equilibrium condition is defined by $PA_{r}=W$, where $P$ is the equilibrium pressure. Consider the unloaded liquid spring, in which the charging pressure is $P_{c}$ and the piston rests at one end. When the load $W$ is applied, and $P_{c}<P$, then the piston moves down (or the cylinder moves up, depending upon whether the piston or the cylinder is fixed) so that the resultant volume of fluid inside the cylinder reduces and thereby increases the pressure to $P$ from $P_{c}$. The amount of required displacement, $y$, is obtained from

$$\begin{aligned} P=\beta \left( 1-\frac{\rho _{0}}{\rho }\right) , \end{aligned}$$

which gives what should be the value of $\rho$, i.e. what should be fluid volume $V$ for a constant charged mass $m$. Note that initial volume is $A_{p}\left( H-t\right)$ and hence

$$\begin{aligned} m=A_{p}\left( H-t\right) \frac{\beta \rho _{0}}{\left( \beta +P_{c}\right) } \end{aligned}$$

(6.60)

The stiffness of the spring is given as

$$\begin{aligned} K=-\frac{\partial F}{\partial x} \end{aligned}$$

(6.61)

If $x$ is the current displacement of the spring given in quasistatic fashion then the net restoring force acting on the piston is

$$\begin{aligned} F=PA_{r}=-\beta A_{r}\left( \frac{V_{0}-V}{V_{0}}\right) . \end{aligned}$$

(6.62)

For any additional displacement $dx$, the volume change is $\Delta V=A_{r}dx$ which results in restoring force

$$\begin{aligned} F+dF=-\beta A_{r}\left( \frac{V_{0}-\left( V-A_{r}dx\right) }{V_{0}}\right) =-\beta A_{r}\left( \frac{V_{0}-V}{V_{0}}\right) -\beta A_{r}\left( \frac{ A_{r}dx}{V_{0}}\right) . \end{aligned}$$

(6.63)

Thus,

$$\begin{aligned} K=-\frac{\partial F}{\partial x}=-\frac{\beta A_{r}^{2}}{V_{0}}. \end{aligned}$$

(6.64)

Equation 6.64 indicates that the stiffness of the liquid spring can be changed on the fly by varying the fluids bulk modulus or the volume being strained. This insight will be used later to develop a semi-active suspension.

2.1.3 Bond Graph Model of Elementary Liquid Spring

The bond graph model of the isothermal liquid spring is given in Fig. 6.37a, in which the relations for C-fields are governed by Eq. 6.52, isotropic and anisotropic pressures are denoted with subscripts $i$ and $d$, respectively. Each two-port C-field receives two flow informations, $f=\left[ \begin{array}{cc} \dot{m}&\dot{V} \end{array} \right] ^{T}$ from which it calculates to effort variables $e=\left[ \begin{array}{cc} P&P \end{array} \right] ^{T}=\varPhi _{C}\left( \mathop {\displaystyle \int }\dot{m}dt,\mathop {\displaystyle \int }\dot{V}dt\right)$, where function $\varPhi _{C}(.)$ represents Eq. 6.52. The R-elements on the left and right side of the model in Fig. 6.37a represent strain rate dependent damping according to Eq. 6.58. The TF-elements model the resultant force acting on the piston due to the pressure on both sides of the piston according to Eq. 6.59.

The flow through the orifices is modelled by R$_{\mathrm{n}}$ element in the bond graph model. The wall friction is modelled by R$_{\mathrm{p}}$ element, the input excitation is given by Sf:$V(t)$ element and the I-element models the mass of the piston.

Fig. 6.37

a Isothermal liquid-shock absorber model and b its compact form

Note that the flow through the orifice is assumed to be dependent on only the difference of isotropic pressures in the two control volumes. However, the exact volume being strained is less than that contained in the control volume. On the expanding side, some volume is being filled by the liquid flowing through orifices whereas on the compressing side, some liquid is being squeezed out through the orifice. Therefore, considering the effective volumetric strain rate,

$$\begin{aligned} P_{d}=-\mu _{1}\left( \frac{\rho _{p}\dot{V}_{p}-\dot{m}_{p}}{\rho _{p}V_{p}} \right) =-\mu _{1}\left( \frac{\rho _{p}\dot{V}_{p}-\left( \rho _{p}\dot{V} _{p}+\dot{\rho }_{p}V_{p}\right) }{\rho _{p}V_{p}}\right) =\mu _{1}\left( \frac{\dot{\rho }_{p}}{\rho _{p}}\right) . \end{aligned}$$

(6.65)

Note that for a control volume with no mass transfer ($m_{p}=\rho _{p}V_{p}=$ constant),

$$\begin{aligned} \frac{\dot{\rho }_{p}}{\rho _{p}}=-\frac{\dot{V}_{p}}{V_{p}}, \end{aligned}$$

i.e. Eqs. 6.58 and 6.65 give identical results. Furthermore, it is assumed that the shear flow losses due to the coefficient of viscosity, $\mu$, and wall friction losses in the orifice are included in evaluating the orifice discharge coefficient, C$_{d}$, which is usually a nonlinear function of the average coefficient of viscosity and the fluid density. Then the isothermal liquid spring model considering the strain relaxation due to orifice flow is given in Fig. 6.37b, where the RC-field elements are used to represent the combined pressure given by Eqs. 6.52 and 6.65. The constitutive relation for the RC-fields is given by

$$\begin{aligned} P_{i}=\beta \left( 1-\frac{\rho _{0}}{\rho }\right) +\mu _{1}\left( \frac{ \dot{\rho }_{p}}{\rho _{p}}\right) =\beta \left( 1-\frac{V_{i}\rho _{0}}{m_{i} }\right) -\mu _{1}\left( \frac{\dot{V}_{i}}{V_{i}}\right) +\mu _{1}\left( \frac{\dot{m}_{i}}{m_{i}}\right) , \end{aligned}$$

(6.66)

where $m_{i}$ and $V_{i}$ are state variables and $i=1,2$ represents the two control volumes. In Eq. 6.66, rate of states appearing in the right-hand side, which represent damping, lead to an implicit form and consequently an appropriate integration scheme is required for simulation.

The flow through the orifices, modelled by R$_{\mathrm{n}}$ element, is assumed to be governed by Bernoulli damping relation. It leads to a flow characteristics

$$\begin{aligned} f=C_{d}\sqrt{\left|\Delta P\right|}\text{ sign}(\Delta P), \end{aligned}$$

(6.67)

where the coefficient of discharge $C_{d}=n.\varPhi _{o}\left( d,L,\rho ,T\right)$, $\varPhi _{o}$ is a non-linear function, $d$ is orifice diameter, $n$ is number of orifices, $L$ is the length of each orifice, $\rho$ is average density of fluid (a constant here), and $T$ is the fluid temperature. There are various expressions for $\varPhi _{o}$ available in literature for sharp orifices and small bore orifices.

2.1.4 Model Initialization

The change in average density of fluid in shock absorber is only due to change in intrusion of the piston rod. Therefore, coefficient of discharge in orifices remains almost constant and the total discharge coefficient including number of orifices and their dimensions can be used in the model.

The model states need to be initialized appropriately before simulation. From the given piston rod area ($A_{r}$), the steady state pressure may be calculated as $P_{0}=W/A_{r}$. Then, one may assume an initial piston position (say, in middle of the suspension) and prescribe the initial volumes in two chambers (states associated with C-fields). Using Eq. 6.52, determine the initial volumetric strain at the given equilibrium pressure $P_{0}$ and then determine charging or initial density, $\rho$, of the fluid as follows:

$$\begin{aligned} \left( \frac{\Delta V}{V_{0}}\right) =\left( 1-\frac{\rho _{0}}{\rho } \right), \end{aligned}$$

where $\rho _{0}$ is the free space density of the fluid. Once the charging fluid density is known, multiply with initial volumes to calculate the initial mass of fluid in each chamber (other states of the C-fields).

2.1.5 Damping Characteristic

The bond graph model given in Fig. 6.37b is modified to study the damping characteristics of the liquid-spring. First of all, the ground excitation is set to zero, i.e., $V(t)=0$. Thereafter, the load inertia (I:$W/g$) is substituted by a source of flow. The cyclic displacement is given as $x=A\sin (\omega t)$. Taking derivative with time, we get the expression for the source of flow as $\dot{x}=A\omega \cos (\omega t)$. This cyclic velocity is given at the source of flow which substituted the load inertia.

The parameter values chosen for the liquid spring are as follows: piston diameter is 10 cm, piston rod diameter is 5 cm, internal height of the spring (excluding piston height) is 80 cm, orifice discharge coefficient is $10^{-2}$ kg.Pa$^{-1/2}$, $\beta =$10$^{9}$  Pa, and the fluid density is 970 kg/m$^{3}$.

We consider $\pm$10 cm displacement ($A=0.1$ m) cyclic loading at various frequencies to evaluate damping performance. The results for 2cst viscosity silicone fluid are given in Fig. 6.38. The stiffness characteristics is obtained by choosing a very small value of frequency, $\omega =0.01$ rad/s. This is a quasi-static analysis where damping effects are almost absent. The stiffness characteristics is found to be linear within the considered range of deflections. As the frequency of excitation increases, more and more energy (the area enclosed by the hysteresis curve) is dissipated by the damper. Note that the energy dissipated for positive velocities and negative velocities is not the same because of the asymmetry in volumetric strain in the two control volumes.

Fig. 6.38

Liquid spring damping characteristic

2.2 Active Suspensions

Active suspensions use separate actuators and sensors for each wheel. Usually, these suspensions consume great deal of power to provide near perfect shock isolation. The actuators provide independent force on the suspension to improve the riding/handling characteristics. An active or semi-active suspension must perform the following tasks: (a) support the vehicle body at the desired ride height, (b) minimize vertical shock transfer to the sprung mass, (c) correct the suspension level when the vehicle is cornering during a curve negotiation, and (d) provide pitch control when the vehicle accelerates or brakes. For example, when the front wheels of the vehicle go over a bump, the actuators on the front suspension act in such a way that the vehicle does not heave or pitch; the distance between the axle and the vehicle is changed appropriately by the actuator force. When the rear wheels pass over the same bump, the actuators on the rear suspension act the same way. Modern active suspension designs include self-levelling suspension and height adjustable suspension features. In racing cars operating at high speeds, the vehicle is lowered to improve its aerodynamic performance by using the adjustable suspension feature. Such activities require a properly programmed microprocessor or microcomputer to decide on the state of the vehicle from available measurements.

However, achieving all these also requires significant amount of energy, say to operate the pump continuously in a hydraulic/pneumatic active suspension system. Active suspensions are expensive and are thus found in high-end luxury vehicles. Moreover, they require frequent maintenance and the fuel efficiency of the vehicle is reduced due to the added mass and other complications. However, the pitch, cornering and weight change effects may be accommodated slowly because one has sufficient time at hand. On the other hand, controller response to ground excitation needs to be very fast. Thus, it is useful to separate these tasks and provide a fast acting (and large power consuming) actuator and a slow acting actuator. Further note that the large power consuming fast acting actuator (if electrical) can be made regenerative, i.e., generate some power to compensate for the power consumed at other times.

Dampers maintain contact between the wheel with the road by preventing the wheel from continuing upward motion at the crest of a bump. The stiffness and damping of the suspension must be properly matched to obtain the desired suspension response. If damper is offers too little damping then the suspension continues to bounce long after the actual ground excitation. On the other hand, too much suspension damping causes the damper to act as a rigid strut to high-frequency excitations and thus transmits the shock input to the vehicle body which reduces the ride comfort.

In a high bandwidth or fully active suspension system (See Fig. 6.39), an actuator is connected between the sprung and unsprung masses of the vehicle. Such a system aims to control the suspension over the full bandwidth of the system, which may be broadly divided into two resonant frequency ranges of a typical vehicle, namely the rattle-space frequency (10–12 Hz) and tire-hop frequency (3–4 Hz). The fully active suspension system requires actuators with a wide bandwidth which consume large amounts of power. When low band-width actuators (typically around the rattle space frequency) are used, the actuator is placed in series with the spring and the damper shown in Fig. 6.39. Such systems are known as slow-active or band-limited suspensions. Under high frequency input (corresponding to wheel-hop frequency), the actuator acts as a rigid strut and reduces the system to a stiff passive suspension. Thus, one achieves significant reduction in body roll and pitch during cornering and braking manoeuvres. Moreover, the energy consumption is lower than a high bandwidth system.

Fig. 6.39

Schematic representation of an active suspension system

2.2.1 Hydraulic Active Suspension

Hydraulic servo-mechanisms are often used to control active suspensions. Hydraulic power generated by a pump is easily distributed to the suspensions from a common source. On board sensors continually monitor the vehicle motion (ride level) and a computer or microprocessor operates the hydraulic servos of each wheel’s suspension. This feedback mechanism generates suspension forces to regulate body lean, dive, and squat during various driving maneuvers.

The power required to drive the pump reduces the overall fuel efficiency of the vehicle. Moreover, hydraulic systems have slow response and thus the feedback system may become unstable under certain conditions.

2.2.2 Regenerative Electromagnetic Suspension

Electronically controlled active suspension system (ECASS) technology has been developed by L-3 Electronic Systems based on the 1990s design made at the University of Texas Center for Electromechanics. The ECASS-equipped vehicle provides exceptional performance in terms of absorbed power (comfort) and directional stability during tough maneuvering conditions. In this design, linear electromagnetic motors are used in place of hydraulic cylinders. Electromagnetic motors have fast response time than conventional fluid-based damper suspension systems and thus produce smooth variation of suspension force. This quick response aids to virtually eliminate all unwanted movement the body of a car such as roll elimination by automatic stiffening of the suspension when cornering. Therefore, ECASS technology gives the driver a greater sense of control.

Moreover, when the suspension is withdrawing power, the motor can be used as a generator (See discussions on regenerative braking in later sections of this chapter). The power produced through the regeneration action is stored in the battery pack, which improves the overall fuel efficiency of the vehicle.

2.2.3 MR-Fluid Damper

Active hydraulic suspensions implement some kind of control over the suspension stiffness and damping. As far as liquid springs are concerned, they use Silicone fluid which is also a good carrier for synthesis of Magneto-Rheological fluids (MR-fluids). In MR-fluids with Silicone base, variable percentage of micron sized iron particles coated with anticoagulants is used as an additive. The viscosity of the MR-fluid and consequently the fluid flow through orifice is controlled through actuation of magnetic fields in semi-active liquid-spring suspensions. In some recent designs, the damper has a cylinder containing an MR fluid and a sliding piston assembly with a number of concentric annular flow gaps formed between concentrically mounted flux-rings positioned on the piston core. The sliding piston forms a chamber at either end of the assembly. Information from different sensors measuring suspension extension, steering angle, vehicle acceleration and road profile is used to calculate the optimum stiffness for the instantaneous state. The damper increases the overall MR damping and turn-up ratio for a given piston size by utilizing multiple flow gaps. Although the direction of the suspension force cannot be controlled by using MR-fluids, the heavy power requirement to sustain the magnetic field means such dampers fall in the active suspension category.

2.2.4 Design Guidelines

In [38], Karnopp reasons that the conventional damping element between the sprung and unsprung masses in a conventional suspension is in the wrong place because it produces a suspension force depending upon the relative velocities between the sprung and unsprung masses. An optimal suspension should apply a damping force on the sprung mass proportional to the velocity of the sprung mass. The control architecture of the suspension system is shown in the quarter car model given in Fig. 6.40a. In the quarter car suspension model, one usually assumes a constant linear vehicle velocity $\dot{x}$ and from the road profile, the vertical excitation is calculated as $V(t)=\dot{y}=\dot{x}\dfrac{dy}{dx}$. An equivalent passive configuration of the controlled system in Fig. 6.40a is shown in Fig. 6.40b where a damper attached to the inertial frame applies the feedback force. However, in reality, when one end of the damper is attached to the vehicle body (sprung mass), there is no place to attach the other end of the damper to the inertial frame with zero inertial velocity or the ground. Thus, such a conceptual damper is termed a sky-hook damper.

Fig. 6.40

Concept of shy-hook damper

The sky-hook damper is just a conceptual notion; it means that an equivalent force resulting from this conceptual damper should be applied on the sprung mass. However, the force cannot be supplied by an external actuator; the actuator itself must be carried by the vehicle. Thus, if the actuator is an integral part of the vehicle, it has to be mounted between the sprung and unsprung masses. This actuator generates a damping force proportional to the absolute velocity of the sprung mass, but applies it both to the sprung and the unsprung masses (See Fig. 6.41).

Fig. 6.41

Implementation of shy-hook damper

Fig. 6.42

Bond graph model of active suspension system in (a) preferred integral causality and (b) preferred derivative causality

The bond graph model of the quarter car model of the active suspension system is shown in Fig. 6.42a. The bond graph model has four state variables, one controlled source and one sensor.

We can test the structural controllability of this system by assigning preferred derivative causalities to storage elements while allowing both direct and inverse causalities for controlled sources. The resulting bond graph in preferred derivative causality is shown in Fig. 6.42b. It is found that all storage elements can be assigned derivative causality and hence the system is structurally controllable when the actuation is made between the sprung and unsprung masses. However, note that if the wheel loses contact with the ground, the portion within the dotted lines shown in Fig. 6.42b should be removed. In that case, the I-element representing inertia of the unsprung mass would be integrally causalled and thus the system becomes structurally uncontrollable. Thus, the active suspension system is controllable as long as the wheel remains in contact with the ground and that must be ensured by the controller under normal operation. It is left to the reader to evaluate that the system remains observable when the wheel is in and not in contact with the ground.

The suspension performance undergoes significant improvements with implementation of sky-hook damper concept. Figure 6.43 schematically shows the conventional suspension system with a rigid wheel and a sprung mass, its equivalent mechanical system and its frequency response. The adjustment of damper parameter can change the frequency response from the curve shown in solid lines to the curve shown in dotted lines. However, as amplitudes at low-frequency excitation (most notably at the resonant frequency) reduce, the amplitudes grow for high frequency excitations.

Fig. 6.43

Configuration and frequency response of passive suspension

The schematic representation of the active damper with solid wheels, its equivalent mechanical model with sky-hook damper, and the frequency response are shown in Fig. 6.44. The active damping with sky-hook damper concept greatly reduces the low-frequency amplitudes (including at the resonant frequency) and also the high frequency amplitudes, although this reduction becomes marginal for much higher frequencies.

Fig. 6.44

Configuration and frequency response of active suspension

To compare the performance of fully active and passive suspension systems, the parameter values of the suspension are taken from [17] as follows: $m_{s}=240$ kg, $m_{u}=36$ kg, $k_{s}=16{,}000$ N/m, $c_{s}=980$ Ns/m and $k_{t}=160{,}000$ N/m (See Fig. 6.41). The transfer function between the input velocity and output velocity (of the sprung mass) is found to be

$$\begin{aligned} H(s)= \dfrac{18{,}148s+29{,}629}{s^{4}+31.3s^{3}+4{,}955s^{2}+18{,}148s+296{,}296}. \end{aligned}$$

(6.68)

The fully active system generates a suspension force $F_{s}=\alpha _{s}x+\alpha _{d}\dot{x}-\alpha _{sky}x_{s}$ where $x=\left( x_{s}-x_{u}\right)$, $x_{s}$ is the deflection of the sprung mass, $x_{u}$ is the deflection of the unsprung mass, $\alpha _{s}$ is the stiffness coefficient that leads to a force proportional to relative displacement of sprung and unsprung masses, $\alpha _{d}$ is a direct damping coefficient that leads to relative velocity-dependent force, and $\alpha _{sky}$ is the sky-hook damping coefficient that leads to a force proportional to absolute velocity of the sprung mass. For the fully active suspension with $\alpha _{s}=k_{s}$, $\alpha _{d}=c_{s}$ and $\alpha _{sky}=2{,}000$ Ns/m, the transfer function between the input velocity and velocity of the sprung mass is found to be

$$\begin{aligned} H(s)=\dfrac{18{,}148s+29{,}629}{s^{4}+39.64s^{3}+4{,}955s^{2}+5{,}5185s+29{,}6296}. \end{aligned}$$

(6.69)

Note that the suspension system model has four state variables. However, only three of them (partial state feedback) are used for establishing the suspension feedback force $u=F_{s}$. Chalasani [17] proposed a full state feedback of the form $u=-4{,}800X_{1}-1{,}524X_{2}+1{,}248X_{3}+958X_{4}$ and compared the frequency responses of the fully active and passive suspension systems. Karnopp [36, 38] showed that the sky-hook damper configuration (with the suspension force as a function of three state variables) produced remarkably similar response to the active configuration of [17].The frequency response of passive and active suspension systems are compared in Fig. 6.45. Note that in Fig. 6.45, $G(s)=H(s)s$, which is the transfer function between the ground excitation velocity and acceleration of the sprung mass. It is found that the active suspension suppresses the shock transmission around the dominant resonant frequency.

Fig. 6.45

Frequency response for acceleration of sprung mass

The frequency response in Fig. 6.46 shows the shock velocity transmission to the sprung mass. In the active suspension, the input velocity is directly transmitted to the sprung mass (gain $=$ 1) in the complete low frequency range and there is significant attenuation immediately after the first resonant frequency of the corresponding passive system. It shows that the controller developed from a physical idea (sky-hook concept) gives as good a performance as a full-state feedback system developed from pole-placement.

Fig. 6.46

Frequency response for velocity of sprung mass

Based on these ideas, Karnopp [38] proposed two contrasting versions of active suspensions shown in Fig. 6.47. The controller of the first one utilizes only the sprung mass velocity measurement while that of the second one uses velocity measurements of both the sprung and unsprung masses. The first configuration combines a slow-active or low-bandwidth load leveling system with semi-active force generator for high-frequency content absorption. The second configuration uses a single broadband actuator for absorption of both low-frequency and high-frequency excitations. Karnopp [38] suggests that the first configuration results in more practical systems than the second one.

Fig. 6.47

Two configurations for active suspensions

Fig. 6.48

a Active hydraulic force generation device and b its bond graph model

The fully active second configuration can be achieved by a hydraulic contraption shown in Fig. 6.48a. Note that although this contraption can be used to provide any desired suspension force, the hydraulic actuation of valve positioning means slow response. We will have a further look at the hydraulic actuation later in this section. The bond graph model of the hydraulic actuation is given in Fig. 6.48b. In the bond graph model shown in Fig. 6.48b, the MR-element models the valve. It receives the velocity of the sprung mass as the modulating term and accordingly channels the flow through the valve (see the spool valve and hydraulic actuator models given in the previous chapter).

2.3 Semi-active Suspensions

Semi-active systems can only change the parameters of the shock absorber (e.g., viscous damping coefficient, spring stiffness), and do not add energy to the suspension system. The regulation of the suspension force is through passive interaction, e.g., with a spring or a dashpot. Thus, the line of action of the control force is fixed. Semi-active suspensions are less expensive to design and manufacture. They usually consume negligible amount of energy and thereby do not compromise with the fuel efficiency of the vehicle. Recent developments in semi-active suspension design has narrowed the performance gap between semi-active and fully active suspension systems.

2.3.1 Solenoid/Valve Actuated Semi-active Suspension

The basic semi-active suspension consists of a solenoid valve which manipulates the fluid flow inside the hydraulic damper. The solenoids are computer controlled. The control algorithm is developed by using the Sky-Hook damper technique of Karnopp [38]. It is also possible to control the stiffness of the suspension through a switched connection to an auxiliary volume.

In semi-active suspensions, the full suspension force is not generated by an actuator. Usually, hydraulic semi-active suspensions use control valves to channel hydraulic fluid flow in the proper direction by modulating control valves. The main idea is to modulate the energy dissipation in a passive damper through the sensed variables. The power dissipated by the hydraulic damper is given as $P_{d}=F_{d}\left( \dot{x}_{u}-\dot{x}_{s}\right) =F_{d}V_{r}$ where $V_{r}$ is the relative velocity between the sprung and unsprung masses. For the power dissipation to be positive, which is what the suspension is meant to do, it is required that the instantaneous values of $F_{d}$ and $V_{r}$ must be of the same sign. However, all control systems have some feedback delay and actuation delay. Therefore, it is highly possible that the relative velocity may become of wrong sign than what is required to dissipate power. Thus, hydraulic semi-active suspension systems reduce the active component of force to near zero-value whenever the relative velocity is of wrong sign. Two different control laws are used depending upon whether the command is to generate tension of compression. When tension command is required, the control law should produce near-zero compression for wrong sign (negative values) of relative velocity. Likewise, when compression command is required, the control law should produce near-zero tensile force for wrong sign of velocity. This fact is schematically illustrated in Fig. 6.49. The forces and velocities must lie in the first and third quadrants such that the power is dissipated.

Fig. 6.49

Damper force generation for tension command

Two kinds of damper force generation schemes are shown in Fig. 6.49. In one of the cases, the damping force is independent of relative velocity (except around zero velocity). Such forces can be generated by electromagnetically loaded pressure control valves which open or close at pre-designed pressure differences. The second form considers that the generated damping force is strongly related to the relative velocity. Such force generation may be produced by varying valve openings (effectively, valve resistance) through electromagnetic actuation. Although these two forms of force generation seem radically different from each other, it has been shown [36] that the overall suspension performance remains similar.

It is seen from Fig. 6.49 that if the system is designed to switch at $\dot{x}=0$ (from tension to compression command) then it is possible that the force may change sign and instead of dissipating power, power will be imported into the system. Thus, designing a control system for semi-active hydraulic systems is not so trivial. It is usually advisable to shift the switching point from $\dot{x}=0$. In the literature, one finds numerous examples of force control, resistance control , and switching control (between two resistance laws). These kinds of semi-active systems with limited force regulations have been found to yield comparable performance to fully active suspensions.

There are many configurations to produce controllable forces in semi-active hydraulic suspensions. The principal goal is to dissipate energy as the hydraulic fluid is forced through a controlled valve. Some configurations can use multiple valves. A few basic configurations given in [36] are shown in Fig. 6.50.

Fig. 6.50

Different configurations for generating hydraulic semi-active damper forces

In the first configuration (Fig. 6.50a), check valves regulate flow through a single controlled valve used for both tensile and compressive force generation. The minimum pressure in either cylinders of the damper is the reservoir pressure. The check valve may be an electromagnetic poppet valve used for pressure regulation or a position controlled spool valve. The use of a single valve to control both tensile and compressive damper forces requires the actuator to respond fast enough so that the relative velocity does not change sign in between command generation process and thus import energy into the system. To overcome this issue, the other two configurations were proposed.

The second configuration shown in Fig. 6.50b uses separate control valves for controlling tensile and compressive forces. This configuration has the added advantage that the actuators need not have fast response time; the check valves being fast passive systems provide quick switching around $\dot{x}=0$. The third configuration (Fig. 6.50c) is a step ahead with two more control valves and the control valves used to control tension and compression forces are mechanically coupled. The coupling has a symmetric part (i.e., when one control valve position is increased in tension control part, the corresponding control valve position in compression control part increases) and an anti-symmetric part (i.e., when one control valve position is increased in tension control part, the corresponding control valve position in compression control part decreases). The symmetrically coupled control valves change the slope of the force versus velocity curve (i.e., stiffen or soften the damping) and the anti-symmetrically coupled control valves shift the force versus velocity curve to tension or compression side.

More details on design of active and semi-active suspensions can be consulted in [37, 38]. It is noteworthy that semi-active suspensions, which seem to have many limitations, indeed produce comparable performance to much costlier and resource intensive fully active suspension systems. The so-called adaptive suspension, which uses a single control valve, is discussed next in detail.

2.3.2 Adaptive Suspension

Passive liquid spring suspensions belong to the pure hydraulic category. They differ from other hydraulic suspensions in the fact that a conventional hydraulic fluid is not generally used and separate subsystems are not used to control stiffness and damping. Passive liquid springs for sensitive military equipment use check valves instead of orifices in the piston, which allow flow only when designed preload shock is exceeded, and perform as rigid mountings under normal operations. Thus, the design of passive systems is fixed with respect to specific objectives. On the other hand, active and semi-active suspension technologies provide the ability to change the suspension’s natural frequency in real time, in response to the operating environment and control inputs. Hence, it is possible to produce a suspension with low natural frequency when the ride quality is of primary concern and increase the suspension natural frequency when the operating conditions require it. With the ability to change natural frequency on the fly, the vehicle designer can confront the conflicting requirements of a soft compliant ride and a stable safe-handling vehicle with high-load-capacity.

An adaptive suspension allows reconfiguration of the device to suite required bounce and jounce behavior. One may select a compliant suspension configuration on rough roads, whereas a hard suspension behavior can be selected for high-speed maneuvers on smooth highway roads. The reconfiguration of the suspension components may be automatic or manual.

The components and configuration of a compressible liquid adaptive suspension system are shown in Fig. 6.51. We consider a quarter car model in which $M_{V}/4$ (one-fourth of the vehicle body mass $M_{V}$) is the sprung mass, $M_{ax}$ is the unsprung mass (axle and wheel mass taken together), $K_{t}$ is the tire stiffness, $V_{c1}$ and $V_{c2}$ are volumes of two small surge protection accumulators, and $V_{A}$ is the volume of a properly designed large accumulator. The Rate and Damping valves are ON/OFF type, i.e., two position 2-way valves. The external valve is always open; however, no flow takes place through it as long as the damping valve is closed.

Fig. 6.51

Adaptive liquid spring suspension system

The liquid spring shown in Fig. 6.51 does not have holes or annular gaps in the piston. The damping is achieved by a regulated piston valve which connects the control volumes between the two sides of the piston. The working principle is exactly similar to that of the passive liquid spring.

As we know from the preceding section, the liquid spring’s stiffness depends on the volume of liquid. The total liquid volume is given as

$$\begin{aligned} V=V_{s}+V_{c1}+V_{c2}+V_{A}\delta _{r}, \end{aligned}$$

(6.70)

where $\delta _{r}$ is a binary variable. When the rate valve is open, $\delta _{r}=1$, otherwise $\delta _{r}=0$. Thus, the rate valve is used to govern the stiffness of the suspension system. Note that several auxiliary volumes and rate valves may be used to obtain a wide range of stiffness selection.

The flow between the control volumes is given as

$$\begin{aligned} f=\left( C_{dp}+\delta _{e}C_{de}\right) \sqrt{\left|\Delta P\right|}\text{ sign}(\Delta P), \end{aligned}$$

(6.71)

where $C_{dp}$ is the coefficient of discharge through the piston valve, $C_{dd}$ is the coefficient of discharge through the external damping valve, and $\delta _{e}$ is a binary variable governed by the On/OFF state of the damping valve: $\delta _{e}=1$ when the damping valve is open, otherwise $\delta _{e}=0$.

The bond graph model of the adaptive liquid spring suspension is shown in Fig. 6.52. Two modulated R-elements in the model represent the external valve (R$_{\mathrm{e}}$) and the rate valve (R$_{\mathrm{r}}$). The rate valve must be properly controlled such that the spring stiffness does not vary suddenly because it would then suddenly change the equilibrium position of the piston. The new C-element (C$_{\mathrm{A}}$) models the auxiliary volume. There is no pressure volume flow rate port in this new C-element because it has a constant volume. The bottom part of the model includes representations for the tire stiffness and the unsprung mass.

Fig. 6.52

Bond graph model of adaptive liquid spring suspension

The damping valve in combination with the external and piston valves selects the damping level depending on the stiffness of the system (i.e. whether the rate valve is open or closed). The piston and external valves are used to dissipate energy (i.e. provide damping). The flow-pressure gains of the valve (if the flow is governed by linear relation) or coefficients of discharge, and the pressure differential breakpoints (where $\delta _{e}$ and $\delta _{r}$ are modulated) can be adjusted to achieve desired jounce and bounce. This is accomplished without the fuel-consuming power requirements of active suspension technology, since the compressible liquid spring is always reacting passively to road inputs at whatever stiffness has been selected for the moment. The only power consumed is the small amount required by the stiffness selection control mechanism.

3 Anti-Roll Bar and Ride Height Management

When a vehicle takes a sharp turn at high speed, the inertial and centrifugal forces push down the part of the vehicle on the outside of the turn and lift up the part of the vehicle on the inside of the turn. Under severe conditions, the wheels on the inside of the turn may completely lift up from the road. The body roll or lean causes positive camber of the wheels on the outside of the turn and negative on the inside. Whether the wheels completely lift up or not, the normal reaction distribution on the wheels and camber angles change. These changes lead to loss of traction/braking (road holding) and steering during sharp turns. To avoid this, the suspensions’ roll stiffness should be increased. Increasing roll stiffness of the suspension also increases the stiffness of the suspension in the vertical directions and results in uncomfortable drive. Thus, some sort of mechanism is required which would increase the effective roll stiffness[27] of the suspension while maintaining save level of compliance in vertical direction.

3.1 Passive Anti-Roll Bar

The mechanism in question, called an anti-roll bar, turns out to be very simple. Anti-roll bar, also called stabilizer bar and anti-sway bar, is part of the vehicle suspension system designed to reduce the roll of the vehicle while it takes a sharp turn. Anti-roll bar is normally a U-shaped round bar which connects the two front or two rear suspensions of the vehicle. Some vehicles have anti-roll bars for front and rear wheels. The bar is hinged to the vehicle frame using pivoting rubber mounts such that it can rotate about its own axis. Figure 6.53 shows the scheme of the anti-roll bar and its mounting on the vehicle frame.

Fig. 6.53

Schematic diagram of anti-roll bar and its mounting

When the vehicle takes a turn, the front (and rear) suspension strut on the outside of the turn is pushed up, i.e., it compresses. This applies a torsion moment on the anti-roll bar (the moment or lever arm $d$ is shown in Fig. 6.53) which also pushes down the suspension on the inside of the turn. As a result, the vehicle body remains parallel to the road. In this way, the weight on the wheels of the vehicle is evenly distributed and the vehicle’s tracking performance remains comparable to that while the vehicle runs on a straight path. Improved tracking performance also means comfortable handling and safety during cornering.

Anti-roll bar configuration influences the understeer or oversteer behavior of the vehicle. It is known that the slip angles (the simplest form of tire side force representation) determine whether a vehicle understeers or oversteers. If the cornering coefficient of the front wheels is lower than that of the rear wheels, then the rear wheels stick more to the road and the vehicle under steers. On the other hand, if the cornering coefficients of the front wheels are larger than that of the rear wheels, then under action of the same centrifugal force, the rear wheels slip more to the outside of the turn and the vehicle oversteers. Usually, for better handling of the car, understeer characteristic is required because it gives the driver enough time to adjust steering in one particular direction. If a vehicle oversteers, then this adjustment is uncomfortable because the driver has to turn the steering wheel in reverse directions (clockwise and anticlockwise) in quick succession. The load distribution on the wheels also affects the cornering force. The proportion of the total roll stiffness (from the combined effective stiffness offered by the suspension springs and the anti-roll bar on both the axles) offered by the front and rear axles determines the loads transferred to the inside and outside wheels in a turn. If the roll stiffness on the front wheel is large, then the load distribution on the left and right front wheels would be more even whereas it would be uneven on the rear wheels. Because the linear velocity of the wheel on the inner side of the turn is smaller than that of the wheel on the outer side of the turn, the slip angle for the front wheel on the outer side of turn becomes larger than that of the wheel on the inner side and the vehicle understeers. On the other hand, if the roll stiffness on the front wheel is small compared to that of the rear wheel, the vehicles’ understeer is reduced.

Note that the anti-roll bar is an absolute necessity in any vehicle because without it, the driver will have a lot of difficulty to take a turn. At the same time, if the suspension level equalizing effect is too much then the independence between the suspension members on both sides of the vehicle is lost. For example, if one wheel of the vehicle falls into a pit due to which the suspension on that side extends temporarily, then the anti-roll bar transmits the same effect to the suspension on the other side of the vehicle and results in unwanted extra pitch. Likewise, a wheel hitting a bump causes the wheel on the other side to lift up. Thus, there must be a balance between the level of body roll control offered by the anti-roll bar and independence of the suspensions. It is usually difficult to do much about this issue while using a passive anti-roll bar system. This is why, active and semi-active anti-roll systems have been developed which retain full independence between the suspensions on the two sides of the vehicle.

3.2 Active Anti-Roll and Ride Height Management System

The basic layout of a semi-active anti-roll system developed by Automotive Products is shown in Fig. 6.54. This system has a very fast response time. It effectively eliminates braking dive (forward pitch) and cornering roll by independently controlling all four suspensions. Moreover, this design allows the use of very soft suspension springs and thus, significantly improves the ride comfort. It also provides the self-leveling capability which adjusts the vehicle height upon load change, such as extra passengers boarding or leaving the vehicle.

Fig. 6.54

Schema of anti-roll system developed by Automotive Products

The Automotive Products anti roll system has a damper valve between a gas spring and the piston. There are four such suspension systems, each for a wheel. All of these suspension struts are connected to a common high pressure hydraulic pump, which is driven by the engine. The high pressure reservoir is the same one used to operate other hydraulic systems such as the brakes. Each suspension unit has a ride-height sensing control valve through which fluid flows in or out to adjust the suspension length and thus, vehicle’s ground clearance. The level correction is a slow process, i.e., it has a long response time. This is not a problem because the load change only takes place few times at considerably larger intervals. On the other hand, adjustments to road irregularities and inertia-induced loads (braking, cornering) require quick response. The Automotive Products anti-roll system acts as purely passive suspension (provided by the pneumatic parts) for such fast load changes. The active part (hydraulic part) is only used for level adjustment and thus the power and fast response time requirements are reduced.

The pendulous mass allows the device to determine whether the suspension motion is due to inertial effects (pitch, roll) or due to road unevenness. The inertia effects cause the suspension arm tilt and the spool valve is actuated through the linkages to force hydraulic fluid into/ out of the cylinder. Road unevenness moves the suspension arm vertically and thus the hydraulic part is not actuated. Then the gas spring and the valve damper provide the suspension damping.

In some forms of semi-active suspensions, the power required to pump the hydraulic fluid is generated from wheel movement due to road shocks. To harness the required power through regeneration, the vehicle must be run for some distance. As the vehicle encounters road unevenness, the suspension damping action is provided by charging a high-pressure chamber through fast acting check valves. Thus, the correct ride-height is obtained only when the vehicle has traveled some distance.

A fully active ride height adjustment system uses many sensors and actuators with a microprocessor-based control system. The load on the vehicle, velocity, accelerations in three linear axes, and angular accelerations (pitch and roll) are the sensed variables. In such a design, the vehicle weight is entirely supported by actuator generated forces. These actuators, besides providing compliant ride and good maneuverability, also manage roll, pitch and ride-height variations. However, supporting the entire vehicle on actuators consumes a lot of power. More details on such active suspension systems are already given in the previous sections of this chapter.

4 Power Steering

In a vehicle with manual steering, the driver has to supply the required steering moment to orient the steerable wheels in the desired direction. To overcome the huge force required to turn the wheels, mechanical advantage is provided in the form of steering gear and linkage. Still, the driver has to apply significant effort especially when the vehicle speed is low and the load is heavy.

Modern vehicles being bigger and heavier require huge traction and braking force. Therefore, the contact patch between the wheel and the road needs to be increased. This is why such vehicles use wider and low-pressure tires. However, as the vehicle gains more grip on the ground, it requires more effort to steer it. The huge physical strength required to drive such vehicles with manual steering is an impediment to older, handicapped, and other less-able drivers. Power-assisted steering (PAS) was introduced for this purpose. In PAS, an auxiliary actuator adds on power to the driver supplied power on the steering wheel. The auxiliary actuator may be pneumatic, hydraulic, or electric.

4.1 Drive-by-Wire System

In a drive-by-wire or steer-by-wire power steering system, the auxiliary actuator provides the full force to move the wheels and the driver power is not used in the process (See Fig. 6.55). The actual rotation of the steering wheel and the wheel rotation are measured and compared in a controller which then commands the electric or hydraulic actuator to generate the torque required to steer the wheels. Moreover, the controller also commands a feedback motor to generate a much smaller reaction torque which is felt by the driver. In essence, the driver gets the feeling of being actually able to steer the wheels with much reduced effort. Note that a simple torsion spring does not give the correct feedback moment to the driver. The controller has to take additional factors, such as the load on the wheels and the vehicle speed, into account to calculate the correct moment feedback to the driver.

Fig. 6.55

Drive-by-wire or Steer-by-wire fully active power steering system

4.2 Integral Power Steering

In integral power steering systems, a torsion bar is attached to the end of the worm shaft. When the driver rotates the steering wheel, the torsion bar twists. This twist may be used as a torque sensor. In power assisted hydraulic/pneumatic steering systems, as the torsion bar twists, it moves a spool valve slightly away from its neutral position. This allows fluid flow from a high-pressure fluid reservoir into the hydraulic actuator or power cylinder. The piston in the power cylinder moves and adds power to the steering axle. The power cylinder is connected to the steering linkage in linkage-type power steering systems. In rack and pinion-type power steering systems, the power cylinder is a part of the rack, i.e., the rack motion is due to the force applied by the driver and the hydraulic piston.

The hydraulic cylinder and its control through spool valves have been already discussed in the previous chapter. The only difference is that in the previous chapter the spool valve is actuated by a solenoid with current control whereas in power steering applications, it is moved in proportion to the steering wheel rotation by mechanical means. Note that hydraulic actuators have slow response time. Therefore, electrical actuators are the better option. In this chapter, we will deal with electric power assisted steering (EPAS) systems. Note that the bond graph models for EPAS systems developed in this chapter can be readily changed to hydraulic power assisted systems by suitable replacement of the actuator component model by the hydraulic cylinder and spool valve bond graph models of the previous chapter.

4.3 Differential-Type Power Steering

A simple configuration of an EPAS, as shown in Fig. 6.56a, has been proposed in [47]. This EPAS system uses a differential which allows an electric motor to add power to the power provided by the driver without forcing the wheel from the driver’s hands. Such a system has also been developed in [53].

Fig. 6.56

Schematic representation of (a) differential type power steering system and (b) mechanical trail

The forces resisting the steering motion are the self-aligning moments ($M_{z}$) on the two wheels and the moments created by the cornering forces ($F_{c}$) and the mechanical trail of the contact patch ($\varepsilon$). The contact patch trails the vertical axis of rotation of the wheel as shown in Fig. 6.56b. The mechanical trail is due to the elastic deformation of the tire by hydrodynamic forces (and elastic waves) at tire road interface [50]. The bond graph model of the EPAS system with the differential is shown in Fig. 6.57.

Fig. 6.57

Bond graph model of differential type EPAS

The top left part of the bond graph model represents the torque applied on the steering wheel, and the rotary inertia and damping offered by the steering column. The rotation of the steering wheel is scaled by a reduction wheel. The bottom left part of the model concerns the assistance provided by an electric motor, whose input voltage is modulated depending upon the rotation of the wheel about the vertical axis ($\delta$). This modulation may be done through various means, the most common of them being the scheme for pulse-width-modulation control of geared DC motors. This simple model does not consider separate values of $\delta$ for the two wheels. The differential is represented by the 0-junction. Pacejka’s magic formula is used as the constitutive relation for the MR-element which generates the tire’s self-aligning moment, cornering forces, and other forces as has been discussed during the wheel model development given previously.

4.4 Electric Power-Assisted Steering Model

A more accurate bond graph model of the EPAS system is given in [49]. The schematic representation of the considered EPAS is shown in Fig. 6.58. Such systems are now in use in many hybrid electric vehicles.

Fig. 6.58

Schematic representation of the EPAS system

The EPAS system consists of a principal actuating line (steering wheel, steering column, torsion bar and the pinion), and a secondary actuating line (electric motor, reduction gears, drive column, and another pinion). The bond graph model of this system is shown in Fig. 6.59.

Fig. 6.59

Bond graph model of the EPAS system

The steering wheel torque input, steering wheel and column inertia, and the resistance are modeled at the 1-junction in the top left part of the bond graph model. Likewise, both the pinions are assumed to be of the same size and thus their rotations must be the same. They are modeled at a 1-junction with a combined rotary inertia and damping. The torsion bar is modeled by a C-element representing the stiffness of the torsion bar. The twist of the torsion bar is fed to the controller (not shown in the model) which modulates the voltage supply (MSe-element) to a DC motor. The DC motor model includes coil inductance, resistance, motor characteristics (GY:$\mu$), rotor inertia and damping, and the reduction gear. The model given in [49] also includes transmission efficiency, which is modeled by an MR-element. Here, we use a gear mesh coupling for that purpose. This coupling may be used to simulate backlash in the transmission.

The rack and pinion gear ratio (combined for both the pinions) is modeled by a TF-element. The rack inertia and linear resistance are modeled at a 1-junction and two MTF-elements modulated by the pinion rotation angle (See steering model) are used to couple the rack motion to the motion of the wheels. The modulation law for the two MTF-elements should take into account the steering linkage geometry, camber angle, and steering axle inclination [18]. The wheel model has been already developed in this chapter and it gives the reaction forces to the rack motion. Note that some of the inertias and resistances considered in this detailed model are so small with respect to the stiffness of the coupling elements that they may be lumped together to create a much simplified (or reduced order) model of the EPAS system.

5 Antilock Braking System

The tire, suspension, and brake system interaction is the most important from the point of vehicle safety. Therefore, suspension and brake systems are the major and critical areas of mechatronization.³

From the slip–friction curve discussed previously in this chapter, it is known that coefficient of friction increases with the increase of slip to a maximum value and then decreases to a minimum value. The longitudinal slip ratio is defined as the normalized difference between the circumferential velocity and the translational velocity of the driven wheel,

$$\begin{aligned} \sigma _{ \mathrm{x}}=\left\{ \begin{array}{l} \dfrac{\dot{\theta }_{\mathrm{wy}}r_{\mathrm{w}}-\dot{x}_{\mathrm{w}}}{\dot{\theta } _{\mathrm{wy}}r_{\mathrm{w}}} \ \left( \mathrm{during\; }\; \mathrm{traction,\; assuming\ }\; \dot{\theta }_{\mathrm{wy}}\rangle 0\right) \\ \dfrac{\dot{x}_{\mathrm{w}}-\dot{\theta }_{\mathrm{wy}}r_{\mathrm{w}}}{\dot{x}_{ \mathrm{w}}} \ \left( \mathrm{during }\; \mathrm{braking,\; assuming\ \ }\; \dot{x} _{\mathrm{w}}\rangle 0\right) \end{array} \right. \end{aligned}$$

(6.72)

where $\dot{x}_{\mathrm{w}}$ is the linear speed of the wheel, $\dot{\theta }_{ \mathrm{wy}}$ is its angular speed, and $r_{\mathrm{w}}$ is the radius of the wheel. The tire friction force (or the coefficient of friction) are obtained from the empirical formulas by Pacejka [58] or Burckhardt [14, 15], as has been discussed in this chapter. It is easily seen from the slip-friction curve shown in Fig. 6.60 that if the wheel gets locked$\left( \sigma _{ \mathrm{x}}=1\right)$ then the coefficient of friction becomes small and the wheel starts sliding. Consequently, the steering control is lost, which is totally undesirable. Hence, to increase steerability and lateral stability of the vehicle and to decrease the stopping distance during braking, the slip value must be maintained within a range to get the high value of friction force. As the slip dynamics is very fast and at any value after the peak of the friction curve is open loop unstable, the slip value is kept within a certain range which is also called sweet–spot. The sweet–spot varies for different reasons such as tire type, tire pressure, tire temperature, road conditions.

Fig. 6.60

Friction coefficient versus slip ratio curves for different road surfaces

ABS as a vehicle autonomous system can be used to improve stability and to reduce longitudinal stopping distance while maneuvering under braking condition. Antilock braking system is suitable for dangerous braking conditions such as braking on icy or wet asphalt roads or for panic braking situations. Upon braking, when the wheel starts slipping, i.e., the slip ratio increases to a maximum desired value, the braking torque is to be reduced and consequently, the speed of the wheel increases, i.e., the slip ratio decreases. Again when the slip ratio meets a minimum desired value, the braking torque is increased and the process continues. The control of ABS is a combination of slip and wheel acceleration control. In wheel acceleration control, the wheel angular velocities are measured and slip is controlled indirectly by changing the speed of the wheels. Slip cannot be kept in an acceptable range for conventional ABS. This is the main drawback for this system. Lots of testing and tuning are required for every ABS algorithm.

Some brake controllers receive direct, fast, and continuous information about dynamic events in the tire contact area from a tire sensor, which is generally embedded inside the tire tread and measures tread deformation, tire pressure, tire temperature, etc. The tire sensor output is used to distinguish between straight line driving or cornering and to determine the $\mu$-split conditions (i.e., differential grip on opposing wheels as a function of road condition and normal load transmitted from the suspension). For example, tire sensors can identify if one of the wheel is offering higher grip as opposed to the wheel on the opposite side. This allows independent brake control for each wheel.

The schema of a quarter car model and its bond graph model are shown in Fig. 6.61a,b, respectively. The equations of motion for rotational and linear dynamics of the quarter car model during braking are given by

$$\begin{aligned} \left( m_{ \mathrm{w}}+\dfrac{m_{\mathrm{c}}}{4}\right) \ddot{x}&=-F_{\mathrm{x}^{^{\prime }}}-C_{\mathrm{aero}}\dot{x}^{2} \end{aligned}$$

(6.73)

$$\begin{aligned} J_{\mathrm{wy}}\ddot{\theta }_{\mathrm{wy}}&=F_{\mathrm{x}^{^{\prime }}}r_{\mathrm{w }}-\tau _{\mathrm{b}} \end{aligned}$$

(6.74)

where $m_{\mathrm{w}}$ is the mass of the wheel, $m_{\mathrm{c}}$ is the mass of the car body, $F_{\mathrm{x}^{^{\prime }}}$ is the braking force due to friction, $C_{\mathrm{aero}}$ is the aerodynamic drag coefficient, $J_{\mathrm{wy}}$ is the rotary inertia of wheel and the axle, $\tau _{\mathrm{b}}$ is the brake torque applied on the axle by brake pedal force, and $r_{\mathrm{w} }$ is the radius of the wheel.

Fig. 6.61

a Schema of wheel under braking condition and b its bond graph model

In the bond graph model, the inertia ( $m_{\mathrm{w}}$) of the wheel and the vehicle body ($m_{\mathrm{c}}$) is modeled by I-element at a 1-junction. Similarly, the rotary inertia ($J_{\mathrm{wy}}$) is modeled at another 1-junction. The three-port modulated R-field implements Burckhardt’s formulae, i.e., $F_{\mathrm{x}^{^{\prime }}}=\mu \left( \sigma _{x},\dot{x} _{c}\right) F_{z}$, where the friction coefficient varies as shown in Fig. 6.60. The normal reaction force on the wheel, $F_{\mathrm{z} }=\left( m_{\mathrm{w}}+m_{\mathrm{c}}/4\right) g$, modulates the R-field as an external signal input. The regenerative braking force modeled in the bond graph will be discussed in later sections of this chapter. For the time being, it may be assumed to be zero source of effort.

5.1 Antilock Braking Algorithm

The main components of ABS are brake servo, lever arm, cable, return spring, rod, cam, rotors, and brake pads. The schema of an ABS is shown in Fig. 6.62a. The ABS controller controls the voltage that is fed to the motor. The resistance is in series with the motor. The lever arm is connected to the motor. The cable is connected to the arm at one end and to the lever with cam at the other end. The return spring is used to bring back the brake shoes to their parked state. The flowchart for the ABS algorithm is shown in Fig. 6.62b.

Fig. 6.62

a) Schema of an ABS and b the flow diagram for ABS algorithm

The longitudinal slip is controlled between maximum and minimum values and accordingly brake force signal (and ultimately braking torque) varies as follows:

$$\begin{aligned} \tau =\left\{ \begin{array}{ll} \tau _{\mathrm{previous}}&\mathrm{if }\sigma _{\mathrm{low}}\le \sigma _{\mathrm{x}}\le \sigma _{\mathrm{high}} \\ \tau _{\max }\;\;&\mathrm{if}\sigma _{\mathrm{x}}<\sigma _{ \mathrm{low}} \\ 0&\mathrm{if \ }\sigma _{\mathrm{x}}>\sigma _{ \mathrm{high}} \end{array} \right. \end{aligned}$$

(6.75)

A comparison between the bond graph model of mechanical equivalent braking system (Fig. 6.62a) and the conventional hydraulic braking system is shown in Fig. 6.63. In hydraulic braking system, the pedal force $F_{\mathrm{pedal}}$ is represented by the Se-element (Fig. 6.63b). The fluid flow through supply or hold valve is modeled by a resistive element $R_{\mathrm{sv}}$. The C-element connected with 0 junction, which determines pressure of brake cylinder, indicates the brake fluid compressibility $K_{\beta }$ (a function of bulk modulus and fluid volume). $R_{\mathrm{rv}}$ is the resistance in the pressure relief valve and atmospheric pressure is indicated by the zero-valued Se-element. Braking torque depends on the area of the brake cylinder and the radius of the brake drum.

Fig. 6.63

Bond graph models of (a) mechanical equivalent braking system; and (b) hydraulic braking system

In the bond graph model (Fig. 6.63a) of the mechanical equivalent braking system [42] shown in Fig. 6.62a, the controlled voltage from the ABS controller is fed to the motor. The motor torque ( $\tau _{\mathrm{m}}$) generated is represented by an Se-element. The resistive element connected to the 1-junction denotes mechanical losses ( $R_{\mathrm{lm}}$). The cable stiffness is represented by C-element connected to the 0-junction. The return spring having stiffness $K_{\mathrm{re}}$ is represented by a C-element. The other end of the return spring is anchored to the ground by a zero-valued source of flow. The output braking torque is applied on the wheel.

5.2 Bicycle Vehicle Model

A bicycle model of the vehicle is an improvement over the quarter car model because it includes the steering and cornering dynamics. The bicycle model works perfectly well when the turning radius is large in comparison to the wheel base length. This model does not take into account the roll, pitch, and heave motions and the suspension dynamics is neglected. Thus, the load transfer during maneuvering and braking cannot be included in this model. As the road is considered to be flat, the motion of the vehicle is planar. A schematic of the considered vehicle is shown in Fig. 6.64.

Fig. 6.64

Schema of the bicycle vehicle model

5.2.1 Kinematic Relations

The kinematic relations may be used to construct the backbone of the bond graph model of the bicycle [8]. Thereafter, the wheel rotations and longitudinal and lateral slip calculations may be inserted into the backbone model. It is assumed that only the front wheel is steered (by steering angle $\delta$). The word bond graph of the bicycle model is shown in Fig. 6.65.

Fig. 6.65

Word bond graph of bicycle model

The model has to be developed in non-inertial frame. Note that a steered vehicle (bicycle in this case) is a nonholonomic system, i.e., the constraints on the motion are not integrable. This means, it is not possible to explicitly write down constraint relations as functions of displacements. However, the constraints are expressed in terms of velocities.

Let inertial reference frame be given by X-Y axes in Fig. 6.64 and x-y axes are body fixed (at the center of mass of the bicycle and aligned with the principal axes of the vehicle body). The velocity components along normal and tangential directions to the plane of rotation of the front wheel are

$$\begin{aligned} v_{\mathrm{nfr}}=(\dot{y}+\dot{\theta }_{\mathrm{cz}}a)\cos \delta -\dot{x}\sin \delta \end{aligned}$$

(6.76)

$$\begin{aligned} v_{\mathrm{tfr}}=(\dot{y}+\dot{\theta }_{\mathrm{cz}}a)\sin \delta +\dot{x}\cos \delta \end{aligned}$$

(6.77)

where $\dot{x}$, $\dot{y}$ and $\dot{\theta }_{\mathrm{cz}}$ are the two linear and angular velocities in the body-fixed frame, $a$ and $b$ are geometrical dimensions shown in Fig. 6.64, and $\delta$ is the front wheel steering angle.

Likewise, velocities normal and tangential to the plane of rotation of the rear wheel are

$$\begin{aligned} v_{\mathrm{nrr}}&=(\dot{y}-{\dot{\theta }}_{\mathrm{cz}}b) \nonumber \\ v_{\mathrm{trr}}&=\dot{x} \end{aligned}$$

(6.78)

From Newton-Euler equations (Eqs. 6.5–6.10) with $\dot{z}=\dot{\theta }_{cx}=\dot{\theta }_{cy}=0$, one obtains

$$\begin{aligned} m_{\mathrm{v}}\ddot{x}&=m_{\mathrm{v}}\dot{\theta }_{\mathrm{cz}}\dot{y}+\sum F_{x} \end{aligned}$$

(6.79)

$$\begin{aligned} m_{\mathrm{v}}\ddot{y}&=-m_{\mathrm{v}}\dot{\theta }_{\mathrm{cz}}\dot{x}+\sum F_{y} \end{aligned}$$

(6.80)

where $\sum F_{x}$ and $\sum F_{y}$ are the external forces acting on the vehicle body in respective directions (aerodynamics forces, and cornering and longitudinal forces due to tire–road interaction) and $m_{\mathrm{v}}=m_{\mathrm{c}}+4m_{\mathrm{w}}$.

5.2.2 Bond Graph Model

In the word bond graph of the bicycle vehicle model shown in Fig. 6.65, CTF blocks represent necessary coordinate frame transformations. Maintaining the model structure defined in the word bond graph, the complete bond graph model as shown in Fig. 6.66 can be drawn using Eqs. 6.76–6.80.

Fig. 6.66

Bond graph model of bicycle vehicle model

The vehicle inertia (I$:m_{v}$) is modeled in the moving frame at $1_{\dot{x} }$ and $1_{\dot{y}}$ junctions. The rotary inertia (I$:J_{v}$) is modeled at another 1-junction. The transformer elements (TF) are used to calculate the tangential and normal velocities at tires according to Eqs. 6.76–6.78. The terms $m_{\mathrm{v}}\dot{ \theta }_{\mathrm{cz}}\dot{y}$ and $-m_{\mathrm{v}}\dot{\theta }_{\mathrm{cz}}\dot{x}$ in Eqs. 6.79 and 6.80 are conservative pseudo-forces which can be implemented by a gyrator (GY) element in the bond graph model. The flow detectors (Df) connected to velocity points in the inertial frame are not present in the actual system, i.e., the actual system is not instrumented with an inertial sensor. These flow detectors are simply added to plot the positions of the vehicle center of mass in the inertial frame and also to modulate the MTF elements in the part of the junction structure. The external source of effort (SE:$\tau _{E}$) supplies the engine torque after passing through reduction gears to the rear wheel (differential is not included in bicycle model). The rotary inertia of the two wheels on the rear axle and that of the axle itself ($2J_{w}$) are modeled by I-element at $1_{\dot{\theta } _{wrr}}$ junction. Similarly, the rotary inertia of wheels and axle are modeled at $1_{\dot{\theta }_{wfr}}$ junction. The port number 7 shown within a circle is used to interface this model to the model of the brake system (see Fig. 6.63). The engine is assumed to be fully disengaged from the rear wheel during braking.

The two modulated 3-port R-fields (MR-elements) implement Pacejka’s magic formulae for tangential and cornering tire forces. The normal forces ($F_{z}$) on front and rear wheels are assumed to be constant because roll, pitch, and heave motions due to suspension dynamics are not included in bicycle model. The constant normal forces on the front and rear wheels are $m_{\mathrm{v}}gb/\left( a+b\right)$ and $m_{\mathrm{v}}ga/\left( a+b\right)$, respectively. First of all, the velocity of the contact patch on the tire, the longitudinal velocity of the tire, and the lateral velocity are used to compute the longitudinal and lateral slip ratios. Thereafter, Pacejka’s magic formula is used to compute the longitudinal and lateral tire forces and the reaction moment on the drive. The self-aligning moment is not included in the bicycle model. Thus, the MR-element, as causalled in the model, receives the information of three generalized flow variables, and computes three generalized effort variables.

The CTF block (set of TF-elements modulated by trigonometric functions of $\theta _{cz}$ in Fig. 6.66) transforms velocities in the body-fixed frame to the inertial frame and then the velocities in the inertial frame are integrated to plot the vehicle position as seen by an inertial observer. This transformation is given by

$$\begin{aligned} \left\{ \begin{array}{l} \dot{X} \\ \dot{Y} \end{array} \right\} =\left[ \begin{array}{cc} \sin \theta _{cz}&\cos \theta _{cz} \\ \cos \theta _{cz}&-\sin \theta _{cz} \end{array} \right] \left\{ \begin{array}{l} \dot{x} \\ \dot{y} \end{array} \right\} . \end{aligned}$$

(6.81)

5.3 ABS Performance Simulation

It has been mentioned before that a lot of trial and testing is required before finalizing an ABS controller. Implementation of ABS on the planar bicycle model should be tested to fine-tune and test the ABS control algorithm. Once the controller implementation (program) is found to be satisfactory, it can be ported to a full four-wheel vehicle model. The controller testing may be performed on a simulation model. The model parameters to be used in the simulation of the bicycle vehicle model are given in Table 6.2 where $C_{1}$ to $C_{4}$ are Burckhardt formula parameters. These parameters are used to compute the tire–road friction coefficient $\mu$ during braking and the parameter $D$ of the Pacejka formulae is taken as $D=\mu F_{\mathrm{z}}$. The table lists the nominal value of parameter $D$, which varies according to the slip ratio as the brakes are applied. Subscripts fr and rr are used to identify the front and rear wheels, respectively.

Table 6.2 Parameter values of bicycle vehicle model.

Some simulation and test results for ABS system are available in [42]. Considering the fact that steering effect was not considered in [42], the steering angle in the bicycle model was kept zero. The vehicle weight and the initial linear and angular velocities of the wheels are taken to be the same as those in [42] and the rear wheel of the vehicle is considered to be freely rolling. With these adjustments to match the scenarios, the ABS bicycle model developed in this section is nearly equivalent to that given in [42].

Fig. 6.67

Vehicle speed, wheel speed, and wheel slip ratio as obtained from the bicycle model during full braking by ABS

Figure 6.67 shows the results obtained from the developed ABS bicycle model. One observes almost linear variations in the forward linear wheel speed and a regular pattern of deceleration and acceleration of the wheel angular speed. The slip ratios are initially maintained within a bound and as the vehicle slows down, their maximum values approach unity (i.e., wheel locking). However, the maneuverability is important when the vehicle speed is sufficiently high immediately after panic braking. The low slip ratios during this time allow the driver to steer the vehicle while the brake pedal is still fully pressed.

5.4 ABS Performance While Braking and Maneuvering

The ABS is meant to provide better maneuverability while brakes are applied on the steered wheel. To evaluate the maneuverability of the vehicle under braking situation, a particular scenario has been considered here. First of all, the vehicle starts from rest and it is brought to a steady linear speed of 50 km/h over a straight path. Then it is steered at $t=12$ s to follow a circular path with a constant steering angle of 0.1 radians (about 5.6$^{\circ }$). The steering action causes a decrease in the vehicle linear speed. Once the vehicle linear speed is stabilized and it moves at constant speed over the circular path, front wheel brakes are applied at $t=17$ s. The ABS tries to control the slip ratio between 0.2 and 0.25, i.e., $\sigma _{\mathrm{low}}=$ 0.2 and $\sigma _{\mathrm{high}}=$ 0.25. Note that to handle large side and forward slips during wheel lockup, a composite slip-based formulation [68] developed by the U.S. department of transportation (DOT) has been used to compute tire forces. We encourage the readers to see [68] and appreciate how important it is to correctly model tire–road interaction forces. Note that the results obtained by using Pacejka’s magic formula are not very far away from the results obtained by using composite slip-based formulation.

Figure 6.68a shows the predefined path (2.5 m wide lane) over which the vehicle is supposed to move and the actual paths taken by the vehicle under two different braking conditions: the first when ABS is used and the second when a conventional mechanical brake is used. In the ABS, slip control strategy to keep the slip ratios between 0.2–0.25 (i.e., the sweet-spot) is applied. Initially, a sustained brake force is applied till the slip ratio becomes 0.2 and then the slip control algorithm takes over. The result shows that the vehicle veers off from the lane due to application of the conventional brake, whereas it stays within the lane till it stops when ABS is used.

Fig. 6.68

a Vehicle center of mass trajectory in inertial frame; b wheel slip ratio; c vehicle speed; and d wheel speed for ABS and conventional braking system

The results in Fig. 6.68b show the change in slip ratios due to braking. The slip ratios for ABS remain within 0.2–0.25 and thus the vehicle is able to follow the steering. On the other hand, the slip ratio quickly approaches a value of 1.0 due to application of conventional brake. This causes the wheels to lock up and thus the vehicle cannot steer. Any rotation of the vehicle after the steered wheels are locked is purely due to the initial yaw momentum at the time of braking.

Figure 6.68c shows the vehicle speed variations due to steering and braking and Fig. 6.68d shows the angular velocity of the front wheel under the same condition. The wheel locks up due to application of conventional brake as soon as the wheel speed becomes zero. However, on application of ABS there is an initial sudden drop in wheel speed (till the slip ratio becomes 0.2) and then the wheel speed decelerates and accelerates much like the results shown in Fig. 6.67. The linear speed of the vehicle reduces almost linearly to zero in both case; the deceleration (see Fig. 6.68c) being slightly faster for ABS compared to the conventional brake. This deceleration almost doubles when brakes are applied both on the front and the rear wheels. With the ABS based on slip control mechanism, the vehicle stops faster and at a shorter distance.

5.5 Sliding Mode ABS Control

Pulsating effect (see wheel angular speed in Fig. 6.68d) is one of the major issues in Anti Lock Braking system while in operation. This leads to passenger discomfort during braking. To reduce this pulsating effect and yet maintain the same performance parameters, sliding mode ABS controllers have been proposed. Sliding Mode Control (SMC) is a high-speed switching feedback control that switches between two values based upon some rule. Sliding-mode ABS controllers are designed knowing the optimal value of the desired slip ratio. Brake pressure is increased, decreased, or held during the operation. A problem of concern here is the lack of direct slip measurements. The goal is to obtain a control algorithm which allows the maximal value of the tire–road friction force to be reached during emergency braking and to maintain the friction level around the sweet-spot in the friction-slip curve without variation.

The sliding surface may described as

$$\begin{aligned} S=\left( \sigma _{ \mathrm{x}}-\sigma _{\mathrm{des}}\right) \end{aligned}$$

(6.82)

where $\sigma _{\mathrm{des}}$ is the desired slip ratio.

We can design the controller based on the quarter car model of the vehicle. Then taking $\dot{x}_{\mathrm{w}}\cong \dot{x}$, differentiation of Eq. 6.72 with respect to time during braking period gives

$$\begin{aligned} \dot{\sigma }_{\mathrm{x}}=\dfrac{\dot{x}_{\mathrm{w}}\left( \ddot{x}_{\mathrm{w}}- \ddot{\theta }_{\mathrm{w}}r_{\mathrm{w}}\right) -\left( \dot{x}_{\mathrm{w}}-\dot{ \theta }_{\mathrm{w}}r_{\mathrm{w}}\right) \ddot{x}_{\mathrm{w}}}{\dot{x}_{\mathrm{w} }^{2}}=\dfrac{r_{\mathrm{w}}\left( \dot{\theta }_{\mathrm{w}}\ddot{x}_{\mathrm{w}}- \ddot{\theta }_{\mathrm{w}}\dot{x}_{\mathrm{w}}\right) }{\dot{x}_{\mathrm{w}}^{2}}. \end{aligned}$$

(6.83)

Substituting expressions for $\ddot{x}_{\mathrm{w}}$ and $\ddot{\theta }_{\mathrm{w}}$ from Eqs. 6.73 and 6.74 in the above equation gives

$$\begin{aligned} \dot{\sigma }_{\mathrm{x}}=-\dfrac{1}{\dot{x}_{\mathrm{w}}}\left[ \dfrac{\left( F_{\mathrm{x^{\prime }}}+C_{\mathrm{aero}}\dot{x}_{\mathrm{w}}^{2}\right) \left( 1-\sigma _{ \mathrm{x}}\right) }{m_{\mathrm{e}}}+\dfrac{F_{\mathrm{x^{\prime }}}r_{\mathrm{w}}^{{2}} }{J_{\mathrm{wy}}}\right] +\dfrac{r_{\mathrm{w}}\tau _{\mathrm{b}}}{\dot{x}_{\mathrm{w}}J_{\mathrm{wy}}} \nonumber \\ \quad \,=f\left( \dot{\theta }_{\mathrm{wy}},\dot{x}_{\mathrm{w}}\right) +g\left( \dot{x}_{\mathrm{w}}\right) \tau _{\mathrm{b}}. \end{aligned}$$

(6.84)

To evaluate stability, the Lyapunov function can be written as

$$\begin{aligned} V_{\mathrm{L}}=\dfrac{S^{2}}{2}=\frac{\left( \sigma _{\mathrm{x}}-\sigma _{\mathrm{des }}\right) ^{2}}{2}. \end{aligned}$$

For stability of the sliding motion, the first derivative of the Lyapunov function has to be negative definite and the second derivative must be positive definite. $\dot{V}_{\mathrm{L}}$ is negative definite iff

$$\begin{aligned} \left[ f\left( \dot{\theta }_{\mathrm{wy}},\dot{x}_{\mathrm{w}}\right) +g\left( \dot{x}_{\mathrm{w}}\right) \tau _{\mathrm{b}}\right] \left\{ \begin{array}{l} <0\;\;\forall S>0, \\ =0\;\;\forall S=0, \\ >0\;\;\forall S<0. \end{array} \right. \end{aligned}$$

(6.85)

which is satisfied if a controller gain $\eta >0$ is defined such that

$$\begin{aligned} \dot{S}=-\eta .\mathop {\mathrm{sgn}}\nolimits \left( S\right) \end{aligned}$$

(6.86)

where sgn(.) is the signum function.

Using Eqs. 6.85 and 6.86, and noting that $F_{\mathrm{x^{\prime } }}=\mu (\sigma _{\mathrm{x}},\dot{x}_{\mathrm{c}})F_{\mathrm{z}}$, $m_{\mathrm{e} }=m_{\mathrm{c}}/4+m_{\mathrm{w}}$ and $\dot{x}_{\mathrm{c}}\cong \dot{x}_{\mathrm{w} }$ (as the suspension stiffness in the longitudinal direction is very high),

$$\begin{aligned} \tau _{\mathrm{b}}&=\dfrac{J_{\mathrm{wy}}}{r_{\mathrm{w}}}\left[ \dfrac{\left( \mu (\sigma _{\mathrm{x}},\dot{x}_{\mathrm{c}})F_{\mathrm{z}}+C_{\mathrm{aero}}\dot{x}_{ \mathrm{w}}^{2}\right) \left( 1-\sigma _{\mathrm{x}}\right) }{m_{\mathrm{e}}}+ \dfrac{\mu (\sigma _{\mathrm{x}},\dot{x}_{\mathrm{c}})F_{\mathrm{z}}r_{\mathrm{w}}^{ {2}}}{J_{\mathrm{wy}}}\right] -\dfrac{J_{\mathrm{wy}}}{r_{\mathrm{w}}}\eta \dot{x}_{\mathrm{c}}\mathop {\mathrm{sgn}}\nolimits \left( S\right) \nonumber \\&=q\left( \dot{\theta }_{\mathrm{wy}},\dot{x}_{\mathrm{c}}\right) -k_{\mathrm{g}} \dot{x}_{\mathrm{c}}\mathop {\mathrm{sgn}}\nolimits \left( S\right) , \end{aligned}$$

(6.87)

where $k_{g}=\eta J_{\mathrm{wy}}/r_{\mathrm{w}}$. Sliding motion occurs when states $\left(\dot{\theta }_{\mathrm{wy}},\dot{x}_{\mathrm{c}}\right)$ reach the sliding surface. Equation 6.87 defines the simplest sliding mode controller which calculates the braking torque to be applied on the axle so that the desired slip ratio is maintained at the tire–road interface.

A sliding mode ABS controller with integral feedback has been proposed very recently in [51]. Although the sliding mode controller developed therein has a different structure, it can be used to validate the controller developed in this section. A quarter car model with the wheel weight and one-fourth of the vehicle weight as the load on the wheel is used in [51] to test the controller. Moreover, Dugoff’s tire model is used therein with dry asphalt road condition and the wheel slip is maintained at 0.15 on the sliding surface.

By considering zero steering angle, half-vehicle weight and freely rolling rear wheel, the bicycle model developed in this chapter becomes equivalent to that of a quarter car model. The vehicle weight and the initial linear and angular velocities of the wheels are taken to be the same as those in [51] and the parameters reported in [51] were used to simulate the bicycle model of the vehicle. In [51], the initial linear velocity is taken to be 20 m/s (72 km/h) and the results are plotted for 1.7 s duration. The same settings have been used in our simulations. An initial brake force is applied until the slip ratio reaches 0.15 and then SMC takes over. A comparison of the simulation results is given in Fig. 6.69.

Fig. 6.69

Comparison of sliding mode ABS controller results in [51] with the present model

The result shows that the vehicle linear velocity reduces linearly. After initial increase in slip ratio due to mechanical braking action, the SMC controls the slip ratio at the desired value of 0.15. As a consequence, after the initial drop in wheel angular speed due to mechanical braking, the wheel angular speed reduces linearly. Thus, the passenger feels a smooth braking action instead of the pulsating effect felt due to standard ABS application.

6 Regenerative Braking System

With mechanical friction brakes, the kinetic energy turns into heat energy and gets dissipated into the environment. Hybridization of vehicle driveline has led to the possibility of accumulating energy due to decrease in vehicle’s momentum during braking and consequently make use of that accumulated energy to drive/accelerate vehicle whenever it is intended. This type of a braking process is called regenerative braking. Regenerative braking improves vehicle’s fuel economy, especially in city driving conditions, where frequent braking is required. To apply regenerative braking, an algorithm has to be designed to distribute the braking force into regenerative braking force and frictional braking force.

The main drawback of regenerative braking is that the generated electricity must be closely matched with the supply. Thus, the voltage developed by the generator must be closely controlled. By comparing the estimated regenerative brake torque with the maximum braking torque required to stop the vehicle, decisions are taken on the use of the type of braking in different situations. A continuously variable transmission (CVT), which is an automatic, step-less, smooth transmission system from which infinite number of gear ratios can be obtained within the limits (Fig. 6.70), is required to maintain constant speed at the generator input for various output velocities during regenerative braking process. Alternatively, a power-split device may be used. The regenerative braking force is usually insufficient during panic braking which is why additional braking force has to be provided by mechanical brakes. Pure mechanical brakes can lock up the wheels and thus compromise vehicles’ directional stability. Thus, to ensure quick braking while maintaining vehicle maneuverability, the mechanical braking part can be performed by an ABS.

Fig. 6.70

Schematic representation of a cone and belt CVT

Note that standard ABS schemes cannot be used in conjunction with regenerative braking because that would require fast switching of the CVT ratio during braking and reduce the life of CVT. However, a sliding mode ABS control can be interfaced in parallel with regenerative braking and thus the advantages offered by both can be harnessed.

Fig. 6.71

Schema of regenerative braking

The schema of regenerative braking system is shown in Fig. 6.71. An electric vehicle setup is considered for this analysis on regenerative braking. The vehicle is considered to have rear wheel drives. The drive torque is applied to the differential through motor, CVT, and final reduction gear. The electric motor itself acts as the generator during regenerative braking. The semiconductor switch in the transmission line behaves like a clutch. CVT is used to maintain the generator input speed constant during regeneration. During regenerative braking, the vehicle uses the motor as generator and its output is used to charge the regenerative battery. The braking of the vehicle is performed by using both regenerative and ABS for the present study. Pedal input, vehicle’s longitudinal velocity and CVT ratio are used as input for calculation of regenerative braking and maximum braking torque. When the brake is applied, the controller unit calculates the required torque and its distribution.

6.1 Regenerative Braking Algorithm

An algorithm is required to decide on how to distribute the braking force between regenerative braking and friction or antilock braking in an emergency braking situation. The emergency braking is differentiated from normal or slow braking from the brake pedal position or force. The flow diagram as shown in Fig. 6.72 is used here to decide on distribution of braking forces depending on various input parameters. If the regenerative braking force $F_{ \mathrm{reg}}$ is less than the required maximum braking force $F_{\mathrm{bf}}$ then both regenerative and antilock braking will work in union. If regenerative braking force $F_{\mathrm{reg}}$ is more than the required maximum braking force $F_{\mathrm{bf}}$, then regenerative braking alone will carry out the job.

Fig. 6.72

Flow diagram of regenerative braking

In the present study, the regenerative braking torque applied at the wheels with final reduction gear ratio G is calculated as

$$\begin{aligned} T_{\mathrm{w}}=iGT_{\mathrm{reg}}W=F_{\mathrm{reg}}r_{\mathrm{w}} \end{aligned}$$

(6.88)

where $W$ is the weight factor for state of charge (SOC) whose value is constant up to a certain percentage of SOC and after that value of SOC it decreases linearly [76]. The value of the regenerative torque $T_{ \mathrm{reg}}$ is a constant determined from the motor characteristics curve. The CVT ratio is modulated as $i=\omega _{\mathrm{m}}r_{\mathrm{w}}/\left( \dot{x }_{\mathrm{v}}G\right)$, where $\omega _{\mathrm{m}}$ is the desired generator speed. The desired generator speed is a fixed value such that the voltage developed is the required charging voltage, e.g., about 13.8–14.4 V charging voltage for a 12 V (12.6 V open-circuit voltage at full charge) battery. The CVT ratio is accordingly modulated to maintain constant input speed at the generator end. When the vehicle is driven at constant speed, the CVT ratio is maintained constant as 0.58 (as per data in [76]) so that it works like a simple reduction gear and during regeneration process, the CVT ratio is varied within the range 0.58–2.47.

Fig. 6.73

Bond Graph model of regenerative braking

The bond graph developed for the regenerative module is shown in Fig. 6.73. Voltage of the main battery, represented by SE-element, is used to drive the vehicle. The semiconductor switch denoted by TF-element ($\mu _{\mathrm{sc}}$) is modulated and it acts like a clutch. This is also used as a means to model the pulse width modulation-based control of input power. The power supply is disconnected ($\mu _{\mathrm{sc}}=0$) when brakes are applied. The R-element at the 1-junction represents armature resistance, which decides the current required for driving the vehicle and for charging the battery in the regeneration process. Here the regenerator-battery is modeled as a bank of high-capacity capacitors represented as a single equivalent capacitor. The motor or generator is modeled as a modulated gyrator. The TF-element used to modulate the CVT ratio ($\mu _{\mathrm{cvt}}$) has a constant value in the forward path (i.e., motor mode) and its value is modulated continuously during regeneration process (generator mode) to maintain constant speed at the generator input. During charging and boosting (i.e., when additional power is required, the capacitor is connected to the main energy source), the charging capacitor or battery is connected to the armature by a semiconductor switch represented by a modulated TF-element ($\mu _{\mathrm{sb}}$). This semiconductor switch also protects the battery against loss of charge (when charging voltage is below a certain limit) and overcharging (when charging voltage is above a certain limit).

6.2 Validation of Regenerative Braking

The model parameters used in the simulation of the quarter vehicle model are taken from [76]. The validation of the regenerative braking has been performed by comparing the results with those reported in [76]. The vehicle weight and the initial linear and angular velocities of the wheels are taken to be the same as those in [76]. With these adjustments to match the scenarios, the quarter vehicle model with regenerative braking system developed here is nearly equivalent to that presented in [76].

Fig. 6.74

Comparison of (a) vehicle speed, (b) CVT ratio, (c) battery SOC and (d) slip ratio for conventional CVT control and CVT REGEN control with the results of [76]

Figure 6.74 shows a comparison of the results reported in [76] with those obtained from the developed regenerative quarter car model. The vehicle speed (Fig. 6.74a) almost matches with the simulation and experimental test results given in [76]. The CVT ratio and battery SOC with conventional CVT control and CVT REGEN control are compared with results from the same source in Fig. 6.74b,c, respectively. The CVT ratio with REGEN control increases much faster than the conventional CVT control process. The battery SOC increases during regeneration process. The battery SOC increases marginally more with CVT REGEN control than that with conventional CVT control.

It is seen from Fig. 6.60 that coefficient of friction between the road and the wheel increases with the increase in slip ratio. Yeo et al. [76] showed that the time taken to stop the vehicle is nearly 13.6 s. As a result of this, the distance covered after applying brake is nearly 74.3 m. The time and stopping distance are higher as the value of slip ratio is only 0.01, i.e., the slip ratio is not controlled and the coefficient of friction value between the road and the wheel is very less. In order to avoid this problem, we consider regenerative antilock braking in the next step.

6.3 Modified Full Vehicle Model

The word bond graph representation of the full vehicle model is shown in Fig. 6.75. Some of the bonds in the word bond graph are multi bonds[10]. In the word bond graph, the global system is decomposed into eight subsystems. These are: vehicle body, suspension, wheel, steering, antilock braking system, differential, drive, and regenerative braking system. The four wheels are connected with the vehicle body through suspensions. The steering, anti roll bar, and ABS are coupled with the axle by scalar bonds. Likewise, scalar bonds connect the differential to the rear wheels and the vehicle body. Body weight and aerodynamic forces are connected to the vehicle body by vector bonds. Scalar bonds connect the main battery and the regenerative battery to the differential through motor/generator, CVT, and reduction gear. CTF blocks [49] indicate coordinate frame transformations.

Fig. 6.75

Word bond graph model of full vehicle with regenerative braking

Following static analysis, the normal loads acting on the front and rear wheels when the vehicle is moving in a straight path are given by

$$\begin{aligned} N_{ \mathrm{fr,}\mathrm{rr}}=\dfrac{m_{\mathrm{c}}}{4}g+m_{\mathrm{w}}g\pm \dfrac{h}{4a} F_{\mathrm{d}} \end{aligned}$$

(6.89)

where, $F_{\mathrm{d}}=m_{\mathrm{c}}\ddot{x}+C_{\mathrm{aero}}\dot{x}^{2}$, $a$ is the half-wheel base length (considering mass center is at middle), $h$ is the height of the mass center, and $C_{\mathrm{aero}}$ is the aerodynamic drag coefficient. In dynamic analysis, the normal reaction at any wheel is obtained from the values of the tire stiffness and deflection.

6.4 Performance of SMC-Based ABS with Regeneration

The sliding mode ABS control produces smoother variations in wheel rotational speed as compared to the conventional ABS system, thereby improving passenger comfort. This also means it is possible to obtain almost constant input speed at the generator input while smoothly varying CVT ratio. This concept was verified through simulation of the quarter car model. Because the wheel speed varies smoothly with SMC-based ABS, the CVT ratio is calculated from the expression $i=\omega _{\mathrm{m}} /\left( \dot{ \theta }_{\mathrm{w}} G\right)$ for all values of slip ratio. The vehicle was brought to a steady longitudinal speed of 25 m/s (90 km/h) and then SMC ABS brake with regeneration was applied to the front wheels. The SMC tries to keep the slip ratio at the optimal value 0.2 while keeping the braking force bounded within an upper and a lower force limits till the sliding surface is reached around 0.04 s after application of brakes. The initial action of SMC ABS is similar to conventional braking till the vehicle slip ratio reaches 0.2 after which the sliding mode controller truly takes over.

The results in Fig. 6.76a show the variation of vehicle speed and wheel slip ratio during braking by combined SMC based ABS and regenerative braking. Slip ratio plot shows that the SMC algorithm is able to maintain the slip ratio at 0.2. It is seen from Eq. 6.89 that there will be no load transfer between the wheels if the term $h$ vanishes. To compare the actual results with those obtained without considering load transfer, the value of $h$ was artificially taken to be zero for the later simulation. It is observed that the vehicle stops 0.4882 s earlier due to load transfer effect than that obtained from a model which neglects load transfer. Figure 6.76b shows the variation of CVT ratio during regeneration process.

Fig. 6.76

a Vehicle speed, wheel speed, slip ratio, b CVT ratio, c battery SOC and d vertical deflection of front suspension for CVT REGEN control during load transfer

The variation of CVT ratio with load transfer consideration is somewhat steeper than that without load transfer consideration. Figure 6.76c shows the variation of battery SOC. The change in battery SOC is less when load transfer is considered. The deflection of front suspension in the vertical direction with and without load transfer is shown in Fig. 6.76d. Note that there is some additional deflection of the suspension even when load transfer is neglected. This is due to the fact that the brake torque applied on the wheels to stop its forward movement is applied in the reverse sense on the vehicle and this reactive couple moment being a free vector is translated to the center of gravity; thereby producing forward pitching of the vehicle. The 4.02 cm initial deflection of the suspension is due to the static load. The load transfer causes extra deflection of 4.2 mm during braking and once the vehicle stops, the suspensions regain their initial configuration.

7 Hybrid Vehicles

A hybrid vehicle uses two or more different power sources. The most used type called hybrid electric vehicle (HEV) combines an internal combustion (IC) engine with electric motors. In hybrid vehicles, the mechanical separation of the motor from the drive train offers various design innovations.

Some pollution-free green vehicles use propulsion by compressed air. An IC engine is used to compress the air when the pressure in the cylinder falls below a certain limit. Because the IC engine is not directly coupled with the drive, it is possible to operate it at full efficiency during compression of the air. Such systems are much more efficient than battery charged hybrid vehicles.

7.1 Classification of Hybrid Vehicles

Hybrid vehicles may be classified according to their power train configuration, fuel type, and mode of operation. In a parallel HEV, the electric motor and the internal combustion engine can individually or together drive the vehicle. The internal combustion engine may be operating on petroleum, diesel, bio-fuels, alcohol, etc. We will not go into the details of fuel types but rather concentrate on the drive train types and operation models.

A full hybrid or strong hybrid vehicle can run by drawing power from just the engine, just the batteries, or from both. Thus, high-capacity battery pack is needed to provide sufficient power for battery only operation. The usual approach while both power sources are driving the vehicle is to use a PSD. Alternatively, automatically controlled clutches may be used so that the appropriate clutch is engaged when power is drawn from one source. However, when there is no power-split device and both the sources are driving the vehicle, the electric motor must run at the same speed as that of the IC engine.

Some HEVs use regenerative braking (and regenerative suspensions) or kinetic energy recovery system (KERS) to store energy in a battery pack and then use that energy for boosting the vehicle speed under demand, such as assist during overtaking. These types of vehicles are termed mild parallel hybrids. Due to lack of sufficient motor power, a mild hybrid vehicle cannot be driven solely by the electric motor.

A series-parallel HEV uses a PSD to draw the power from an IC engine and one or more electric motors in any desired ratio. The IC engine is the primary power source. The power-split device is akin to a planetary gear set. The electric motor acts as a generator to charge the batteries during braking. When vehicle requires a speed boost, the electric motor is operated and engaged to the power-split device. The IC engine can be automatically shut down when the vehicle is not moving (such as at traffic signals) and later it may be restarted by the electric motor. This reduces fuel consumption and emissions.

Series-hybrid vehicles are driven by the electric motor which draws power from a battery pack. Although there is an engine, the vehicle power train has no mechanical connection with it. The engine runs a generator when the charge in the battery pack depletes due to use. Because the series-hybrid vehicle is purely driven by electric energy, the battery must be powerful enough to store and deliver large power on demand. Cyclic charging and discharging batteries reduces their life. Therefore, super-capacitors have been developed and combined with battery bank for use in series-hybrid vehicles. A variant of series-hybrid or parallel-hybrid vehicle is called plug-in hybrid electrical vehicle which has much larger energy storage capacity. The batteries of a plug-in HEV may be charged from the mains electricity supply at home to minimize the fuel consumption by the IC-engine.

The fuel cell hybrid vehicle is an electric vehicle where a fuel cell and a battery pack are together used as the power source. The fuel cell uses hydrogen as fuel. Any other fuel, such as methane, that can be reformed to hydrogen may be used in some fuel cells. More details of fuel cells are given in the previous chapter. The fuel cell voltage and power output varies with the load (current drawn). Thus, it is seldom used to directly drive the electric motors. Instead, it is used to charge the electric battery when it is depleted.

7.2 Power-Split Device

The PSD distributes the power produced by two separate sources. To divide the power efficiently, a planetary gear system is used (Fig. 6.77). The size of PSD used in modern vehicles is typically smaller than a soccer ball. The three basic components of the epicyclic planetary gear are: the central Sun gear, the Planet carrier which holds one or more peripheral planet or pinion gears that mesh with the sun gear, and an outer ring gear with inward-facing teeth that mesh with the planet gears. Each of the gears (or carriers) can rotate in different ways and thus provide a wide range of power options. In hybrid vehicle transmissions, the planetary carrier is connected to the IC engine and rotates the ring gear and transmits power to the sun gear through the planet gears (Fig. 6.78). At the same time, the electrical motor is connected to the ring gear which drives the vehicle. When braking, the ring gear drives the electrical motor as a generator. Series arrangement of these units can be used to increase the gear ratio.

Fig. 6.77

Schematic representation of power-split device (PSD)

The power-split device allows the vehicle to operate like a parallel hybrid because the electric motor can power the car by itself, the IC engine can power the car by itself, or both can power the car together. At the same time, the PSD also allows the car to operate like a series hybrid because the IC engine can operate at its peak efficiency independent of the vehicle speed and the excess power can be used to charge the batteries. The power-split device also behaves like a CVT, as has been discussed previously in the context of regenerative braking. It may also be noted that when the sun gear is held fixed, the electric motor connected to the ring gear transmits power to the planet carrier thereby powering to start the IC-engine, which means the vehicle does not need a separate starter motor.

Fig. 6.78

Dissociation of the drive train: sun gear connected to generator, planet carrier connected to engine, and ring gear connected to motor

The schematic representation of a planetary gear system is shown in Fig. 6.79 where links 1–4, respectively represent the sun gear, planet gear(s), ring gear, and planet carrier or arm.

Fig. 6.79

Schematic representation of the links in the power-split device

From Fig. 6.79, the angular velocities may now be written as [3]

$$\begin{aligned} \omega _{1/4}=\omega _{1}-\omega _{4}, \nonumber \\ \omega _{3/4}=\omega _{3}-\omega _{4}, \end{aligned}$$

(6.90)

where $\omega _\mathrm{i}$ is the absolute angular velocity of the i-th link and $\omega _{i/j}$ is the angular velocity of the i-th link with respect to j-th link. Rearrangement of Eq. 6.90 yields

$$\begin{aligned} \dfrac{\omega _{1}-\omega _{4}}{\omega _{3}-\omega _{4}}=\dfrac{\omega _{1/4} }{\omega _{3/4}}=-\dfrac{N_{3}}{N_{1}}, \end{aligned}$$

(6.91)

where $N_{1}$ is the number of teeth in link 1 and $N_{3}$ is the number of teeth in link 3. Note that the velocity ratios $\omega _{1/4}/\omega _{3/4}=-N_{3}/N_{1}$ because of the reversal in direction of rotation between link 1 and planet gears.

Using subscripts ‘s’ for sun, ‘r’ for ring, ‘p’ for planet gear, and ‘a’ for arm or planet carrier, we can write from Eq. 6.91

$$\begin{aligned} \dfrac{\omega _{s}-\omega _{a}}{\omega _{r}-\omega _{a}}=-\dfrac{N_{r}}{N_{s} }, \nonumber \\ \Rightarrow N_{r}\omega _{r}+N_{s}\omega _{s}=\left( N_{s}+N_{r}\right) \omega _{a}. \end{aligned}$$

(6.92)

Let us define the form factor as

$$\begin{aligned} n=N_{s}/N_{p}. \end{aligned}$$

To ensure that the gears mesh properly, their pitch circle diameters must match properly, i.e., $p_{s}+2p_{p}=p_{r}$, where $p$ indicates pitch circle diameter and subscripts identify the gears. From this geometric constraint, we can write

$$\begin{aligned} N_{s}+2N_{p}=N_{r}. \end{aligned}$$

(6.93)

Combining Eqs. 6.92 and 6.93, we obtain a kinematic constraint

$$\begin{aligned} n\omega _{s}+\left( 2+n\right) \omega _{r}-2\left( 1+n\right) \omega _{a}=0. \end{aligned}$$

(6.94)

This kinematic constraint can be used to construct the bond graph model of the PSD given in Fig. 6.80.

Fig. 6.80

Bond graph model of power split device

In hybrid synergy drive (HSD) technology developed by Toyota motors, power-split devices are arranged to form series and parallel hybrid configurations. These two configurations are schematically illustrated in Fig. 6.81, where sun, planet carrier, and ring gears are marked as S, C, and R respectively. MG1 and MG2 are two electrical motors/generators and IC is the internal combustion engine. The load is transferred to the driven wheels through the differential.

Fig. 6.81

a Toyota serial hybrid system and b Toyota parallel hybrid HSD configuration for four wheel drive system

Fig. 6.82

Bond graph model of the power train in parallel hybrid vehicle configuration

Usually, MG1 is used as a starting boost device and charger for electric batteries. MG2 drives the vehicle. While braking, MG2 acts as generator (regenerative braking). This configuration allows the IC engine to operate at its peak efficiency speed. The third generation Toyota HSD uses Ravigneaux gearset (having a small and a large sun gear, and a common carrier gear with two independent planetary gears) which is also commonly used in automatic transmission systems. The bond graph model of the parallel hybrid configuration (Fig. 6.81b) is given in Fig. 6.82. It combines the models for engine, regenerative braking, PSD, differential (gear box may as well be included), wheels and the vehicle body with suspensions. The model of the series hybrid configuration can be drawn in the same way.

8 Automatic Transmission

Automatic and semi-automatic transmissions free the driver from working on the clutch and gears so that the driver can concentrate more on the traffic. This greatly reduces stress on the driver especially on long drives and congested roads. A semi-automatic transmission is partly a manual transmission in which there is no clutch. It is also known as clutch-less manual transmission or automated manual transmission. In this system, there in no clutch pedal. Sensors, actuators, and microprocessor-based controllers are used to engage the drive with the transmission. The automatic clutching is quick and smooth.

A fully automatic transmission further relieves the driver from the job of changing gears. Such vehicles have only the brake pedal and accelerator pedals. Using sensed inputs from these pedals, gear ratios are automatically changed. The controller automatically selects the gear based on vehicle speed and throttle pedal position. A schematic diagram of the automatic transmission and its basic bond graph model are given in Fig. 6.83.

8.1 Components of Automatic Transmission System

The key feature of an automatic transmission is a set of planetary gear sets. Auxiliary devices are used to engage these planetary gear sets. Those are a torque converter, a set of bands, and wet-plate clutches to lock parts of the gear set, hydraulic systems to control the clutches and bands, and a gear pump to pressurize the hydraulic fluid.

Fig. 6.83

Cut-away of an automatic transmission

To achieve more gear ratio options, compound planetary gears are used. A compound planetary gear set usually has one common ring gear connected to the transmission and two sun gears and two sets of planets on two planet carriers. This combination produces four forward gear ratios and one reverse gear ratio. To achieve these gear ratios, some of the gears have to be held fixed and some are to rotate at same speed. Clutches and bands are used to engage gears so that they can be fixed or connected serially. The bands wrap around sections of the gear train and connect to the housing. The bands are actuated by hydraulic cylinders to shift clutches.

A gear pump supplies fluid to the transmission cooler and the torque converter. A spring loaded valve is operated by a governor. The governor rotates at the same speed as the transmission shaft and thus the valve opening is proportional to the vehicle speed. The fluid flow from the pump to the turbine is thus passively controlled by the vehicle speed.

The transmission line contains several shift valves which supply hydraulic pressure to the clutches and bands to engage each gear. The shift valve determines when to shift from one gear to the next or previous gear. One arrangement of the clutches and bands in the compound planet configuration is shown in Fig. 6.84.

Fig. 6.84

Clutches, bands, and planetary configuration of the automatic transmission system

Electronically controlled transmissions monitor vehicle speed, engine speed, throttle position, pressure on the brake pedal, and accelerator pedal and control variety of peripheral devices such as regenerative and antilock braking system, active suspension system, etc. Usually, a fuzzy logic controller is used to correlate the inputs to outputs in predefined zones.

The torque converter is a fluid coupling which can introduce slip between the engine and the transmission. This allows the engine to operate independently of the transmission. The amount of slip depends on the resistance offered by the fluid coupling. If the resistance is large, there will be no slip whereas if the resistance is small, there will be appreciable slip. This resistance is changed by changing the fluid pressure in the coupling (much like the pressure in a mechanical friction clutch). Conventional fluid couplings are inefficient. Therefore, the torque converter is composed of a centrifugal pump, a turbine and a stator. The centrifugal pump is connected to the flywheel of the engine. As it turns, it draws fluid at its center and this fluid impinges on the blades of the turbine and flows along its blades to exit at turbine center. The turbine is connected to the transmission. The stator blades redirect the fluid returning from the turbine so that it does not enter the centrifugal pump. In this way, the efficiency of the torque converter is increased. When the pump and turbine rotate at the same speed, there is no torque transmission because the relative velocity of impinging jet on turbine blades is zero. Thus, the engine power is wasted. Some vehicles use a lockup clutch which locks the turbine and the pump of the torque converter to eliminate the slip.

8.2 Bond Graph Model of Automatic Transmission

A complete model of the NAVISTAR 4700 series 4x2 rear wheel driven truck, powered by a turbocharged, inter-cooled engine, and equipped with a four speed automatic transmission is given in [44]. This model considers the pitch-plane dynamics of the vehicle with tire forces, aerodynamic loading, suspension dynamics, and slope of the road profile [43]. The global or top-level view of the model is given in Fig. 6.85. The driver sends commands to the engine by varying the throttle/accelerator position and the mode of drive (parking, overdrive, normal$\ldots$). The brake pedal position is also sensed. These commands influence the engine output and the gear selection in the drive train. The driver can also directly influence the vehicle dynamics through steering (which is not included in pitch-plane model) and selection of compliance level (active suspension parameters).

Fig. 6.85

Overall model of vehicle transmission system

Fig. 6.86

Word bond graph model of the drive train

The drive train is composed of the pump, torque converter, planetary gears, transmission shaft and the differential. The word bond graph model of the transmission system is shown in Fig. 6.86. The speed of the transmission shaft is measured. This information along with the measured engine speed and throttle position decide the gear shift logic according to which the transmission ratio of the compound planetary gear is selected. Clutches and bands of the compound planetary gear are actuated to realize the desired transmission ratio.

8.3 Torque Converter Model

The bond graph model of the torque converter is shown in Fig. 6.87. The engine power is controlled by the fuel injection controller and modeled according to [4]. The engine torque is applied on the pump where the rotary inertia of the impeller and the flywheel are modeled together. The RMGY-element couples the pump to the turbine. RMGY is a defined bond graph element which has been used in [44]. The power scaling gyrator (PGY) element defined in [41] is similar to RMGY. This element acts like a modulated gyrator in which power is not conserved (the R in RMGY signifies loss of power). A linear RMGY-element has thus two gyrator moduli ($\mu _{1}$ and $\mu _{2}$) which are defined in its constitutive relation:

$$\begin{aligned} e_{1}&=\mu _{1}f_{1}\,\mathrm{and}\,e_{2}=\mu _{2}f_{2}, \nonumber \\ e_{1}f_{1}&\ne e_{2}f_{2}\,\mathrm{if}\,\mu _{1}\ne \mu _{2}. \end{aligned}$$

(6.95)

Fig. 6.87

Bond graph model of the torque converter

In the torque converter, the stator is connected to the casing by a one-way clutch. The stator design ensures that the torque converter acts as torque multiplication device at low turbine speeds. The fluid coupling applies forward and reactive torques on the turbine and the pump, respectively. These torques are functions of the pump and turbine speeds which is why the coupling is represented by a GY element. The non-power conserving RMGY-element used to model the losses in the fluid coupling has a nonlinear constitutive relation which is adopted from the static modeling methodology proposed in [3, 27, 64]. The parameters in this relation are empirically obtained by curve-fitting the quasi-static experimental data. The curve fitting is performed between the torque ratio and the speed ratio during steady operation. Likewise, the capacity factor is another parameter used in the constitutive relation and it is also curve fitted with the speed ratio as a free variable. The functional relationships thus obtained through curve-fitting are valid for quasi-static operation, i.e., these functions do not model the influence of large transients due to sudden variations in the input (engine speed) or the load.

The capacity factor or the coefficient of absorption in torque [3] is defined as

$$\begin{aligned} K&= \dfrac{\omega _\mathrm{i}}{\sqrt{T_{i}}} \nonumber \\&=\varPhi _{k}\left( \dfrac{\omega _{T}}{\omega _\mathrm{i}}\right) =\varPhi _{k}\left( \bar{\omega }\right) \end{aligned}$$

(6.96)

where $\omega _\mathrm{i}$ is the angular speed of the input shaft (impeller pump or engine speed), $T_{i}$ is the input torque on the impeller, $\omega _{T}$ is the output (turbine or transmission shaft) speed, $\varPhi _{k}$ is the fitting function and the non-dimensional number $\bar{\omega }=\omega _{T}/\omega _\mathrm{i}$ is called the sliding velocity.

The torque ratio[44] is defined as

$$\begin{aligned} \lambda&= \dfrac{T_{T}}{T_{i}} \nonumber \\&=\varPhi _{T}\left( \dfrac{\omega _{T}}{\omega _\mathrm{i}}\right) =\varPhi _{T}\left( \bar{\omega }\right) \end{aligned}$$

(6.97)

where $\varPhi _{T}$ is the fitting function. When the sliding velocity approaches 1, the lock-up clutch interlocks the impeller and turbine. In this case, the torque ratio and sliding velocity are both taken to be 1.

The constitutive relation of the nonlinear RMGY-element [44] representing the fluid coupling is given as follows:

$$\begin{aligned} T_{i}&=\left( \dfrac{\omega _\mathrm{i}}{\varPhi _{K}\left( \bar{\omega }\right) } \right) ^{2} \end{aligned}$$

(6.98)

$$\begin{aligned} T_{T}&=\varPhi _{T}\left( \bar{\omega }\right) \left( \dfrac{\omega _\mathrm{i}}{\varPhi _{K}\left( \bar{\omega }\right) }\right) ^{2} \end{aligned}$$

(6.99)

where the input variables are $\omega _\mathrm{i}$ and $\omega _{T}$ and the computed variables are $T_{i}$ and $T_{T}$. A block diagram representation of the calculations needed in the RMGY-element is shown in Fig. 6.88.

8.4 Gear Shift Logic and Transmission System Model

The bond graph model of the transmission system is shown in Fig. 6.89. The turbine model (see bond graph model of torque converter in Fig. 6.87) is connected to a 1-junction where two modulated R-elements model the losses. One of the R-elements represents power loss incurred in charging the pump and the second one represents the loss due to churning of transmission fluid. The torque loss due to churning depends on the gear number (how many clutches and bands are involved and in what configuration) and is a nonlinear expression with linear and quadratic terms.

Fig. 6.88

Block diagram model of RMGY-element’s constitutive relation

Gear inertias are lumped with inertias of shaft segments. The springs and dampers in parallel model the gear mesh stiffness and damping along with the torsional stiffness and damping of lay shafts. The RMTF-element is a non-power-conserving transformer which includes internal dissipation [44]. The power scaling transformer (PTF) element defined in [41] is similar to RMTF. The name RMTF suggests it is a MTF with resistance which receives a pair of flow and effort information in two bonds to compute the complementary power variables in those bonds. This defined two-port element (see power variables in Fig. 6.89) has a separate torque ratio and speed ratio:

$$\begin{aligned} \omega _{2}=\mu _{\omega }\omega _{1}\quad\mathrm{and}\quad\tau _{1}=\mu _{\tau }\tau _{2}, \end{aligned}$$

(6.100)

where $\mu _{\omega }$ and $\mu _{\tau }$ are two scaling constants termed speed ratio and torque ratio, respectively. If $\tau _{1}\omega _{1}\ge \tau _{2}\omega _{2},$one introduces power loss into the model. The inputs to the RMTF-element are the turbine speed and the load torque and its outputs are the load on the turbine and the speed of the output shaft.

Fig. 6.89

Bond graph model of the transmission system

The power loss takes care of loss of efficiency during gear shifts. In the model developed by [44], it is assumed that the speed ratio varies linearly during a gear shift. The efficiency loss is modeled by varying the torque ratio. Before the gear shift and after completion of the gear shift, the speed and torque ratios are the same, i.e., $\mu _{\omega }=\mu _{\tau }$. During the gear shift, $\mu _{\tau }\le \mu _{\omega }$. A blending function (see Fig. 6.90 for a gear downshift) is thus used to implement this feature. Note that the shift duration is typically less than 0.08 s.

Fig. 6.90

Blending function to account for efficiency loss during gear shift

In automatic transmissions, the decision regarding gear shift is taken according to the driver demand (accelerator pedal position, brake pedal position) and the current speed of the vehicle. Usually, a shift-logic chart (included in the code embedded in the microprocessor) is used to take this decision. One such chart is shown in Fig. 6.91 where solid lines indicate the up-shift and dashed lines indicate downshift margins, respectively, and $\omega _{s}$ and $\omega _{m}$, respectively, represent the output shaft speed and a reference maximum output speed. For example, if the drive shaft connected to the differential rotates at 10 % of the peak speed ($\omega _{s}/\omega _{m}=0.1$) and the throttle position (accelerator pedal) is held fixed at 50 %, which corresponds to point A in Fig. 6.91, then the compound planet is engaged in a way to produce the lowest forward gear ratio (1st gear). If the throttle position is not changed then the transmission ratio is held at 1${st}$ gear till the vehicle accelerates and its speed increases to a value so that $\omega _{s}/\omega _{m}>0.35$ (point B in Fig. 6.91) at which point the gear up-shift from 1st to 2nd gear is automatically effected. A detailed experimental automatic shift logic map may be consulted in [27].

Fig. 6.91

Schematic representation of gear shift logic

The transmission efficiency has also been included in the differential model given in [44]. Like the transmission system, ideal speed reduction and non-ideal torque multiplication based on a predefined gear efficiency map has been considered. Although we do not go through the entire model development process given in [44], it is worth mentioning here that the authors of that paper could validate their model with experimental results and produced a reduced order model (mostly removal of small rotary inertias of gears and gear mesh stiffness and damping) which produced comparable results to the full model. Note that a much complete and accurate model of the drive train components is developed in [3] in library form by using AMESim software, which also supports bond graph modeling.

9 Fuel Cells

The heat engine is highly polluting and responsible for effects such as ozone layer depletion and greenhouse effect. In this context, fuel cells, which are efficient and environmentally friendly power-generating systems that produce electrical energy by combining fuel and oxygen electrochemically, are being used as alternative energy sources.

A battery is an energy storage device, which contains the reactant chemicals. The electrodes in a battery take part in the chemical reaction. A battery must be discarded once the reactants are depleted (unless the battery is rechargeable). On the other hand, a fuel cell is an energy conversion device where the reactants are continuously supplied and the products are continuously removed. The electrodes and electrolyte do not participate in the chemical reaction but they provide the surfaces on which the reactions take place and they also serve as conductors for the electrons and ions. Therefore, a fuel cell is a thermo-electrochemical device, which converts chemical energy from the reaction of a fuel with an oxidant directly and continuously into electrical energy.

The basic components of a general fuel cell are two porous electrodes, i.e., anode and cathode, which are separated by a solid or liquid electrolyte. The electrolyte is impervious to gases. Fuel is supplied to the anode side and air is supplied to the cathode side. The oxidation reaction is made possible by conduction of ions through the electrolyte. Many challenges have to be overcome before its successful implementation of a fuel cell. Many issues regarding suitable materials for the electrolyte, interconnects, gas seals, and electrodes have to be addressed. There are also issues regarding cell stack design and life span improvement that warrant immediate attention. Computer control of fuel cell stack for load variation with minimum response time, better stack design, cyclic endurance, and power conditioner for utility services are other issues involved in a fuel cell design. Developing robust controller for integrated fuel cell systems is also a major challenge.

9.1 Classification of Fuel Cells

There are many types of fuel cells currently under research and development. Fuel cells are classified according to the electrolyte used. The major types of fuel cells are Molten carbonate fuel cells (MCFCs) where electrolyte is a mixture of molten alkali carbonates that conducts carbonate ions, low temperature (80–100$^{\circ }$ C) Proton exchange membrane fuel cells (PEMFCs) where a polymer membrane that conducts protons (or hydrogen ions) is used as an electrolyte, phosphoric acid fuel cells (PAFCs) where phosphoric acid is used as electrolyte that conducts protons, Alkaline fuel cells (AFCs) where the electrolyte is an aqueous solution of alkaline hydroxide which conducts hydroxyl ions and solid oxide fuel cells (SOFCs) where the electrolyte is a ceramic that conducts ions at high temperatures (800–1,000$^{\circ }$ C).

The electrolyte substance is specifically designed so that it is an electrical insulator (electrons cannot pass through it) and specific ions can pass through it (e.g., protons for PEMFC and oxygen ions for SOFC). The electrons freed during ionization are forced to travel through the external load and upon reaching the cathode side, they reunite with the ions which have passed through the electrolyte to complete the reaction.

Table 6.3 Different types of fuel cells and their application areas.

Different types of fuel cells produce different amounts of power. Their suitability to specific applications is governed by the power requirement by the target application and operational constraints. The suitability of different fuel cell types for different application areas are given in Table 6.3. PEMFC and SOFC are the most important types because they can cater to wider ranges of application areas. Thus, we will specifically consider these two fuel cell types in this chapter. Other bond graph models of fuel cells can be consulted in [13].

9.2 Solid Oxide Fuel Cell

SOFC provides considerably high system efficiency in comparison to other fuel cell systems with cogeneration [67]. Expensive catalysts, which are needed in the case of proton-exchange fuel cells (platinum) and most other types of low temperature fuel cells, are not needed in SOFCs. Moreover, light hydrocarbon fuels, such as methane, propane, and butane, can be internally reformed within the anode because of the high operating temperature. SOFCs have a wide variety of applications from use as auxiliary power units in vehicles to stationary power generation with outputs ranging from 100 W to 2 MW. SOFCs are also coupled with gas turbines in order to improve their efficiencies. The SOFC is also used in combined heat and power systems.⁴

Fig. 6.92

Schematic diagram of a single cell of SOFC

An SOFC is made up of four layers, three of which are ceramics (see Fig. 6.92). A single cell consisting of these four layers stacked together is typically only a few millimeters thick. Hundreds of these cells are then stacked together in series to form a stack. The ceramics used in SOFCs do not become electrically and ionically active until they reach very high temperature. The electrolyte represents the media through which ions migrate from one electrode to the other; thus causing a voltage difference between the anode and the cathode, and consequently an electric current through an external load. Yttria stabilized zirconia (YSZ), which is a ceramic material, is usually used as the electrolyte material.

In the cathode side of the SOFC, oxygen molecules can diffuse to the catalyst and combine with free electrons to form negatively charged oxygen ions. The electrolyte membrane is an electrical insulator. Only the negatively charged oxygen ions can conduct through the membrane. The conducted oxygen ions combine with hydrogen on the anode side to form water and liberate the electrons. These electrons in the anode side travel through the external circuit to the cathode side to complete the cycle.

The reaction taking place in a SOFC is given as

$$\begin{aligned} \mathrm{Cathode}&:\mathrm{O}_{\mathrm{2}}+\mathrm{4e} ^{-}\rightarrow \mathrm{2O}^{2-} \nonumber \\ \mathrm{Anode}&:\mathrm{2H}_{2}\mathrm{+2O} ^{2-}\rightarrow \mathrm{2H}_{2}\mathrm{O+4e}^{-}+\mathrm{Heat} \end{aligned}$$

(6.101)

The reaction takes place at the so-called triple phase boundary (TPB), where electrons, ions, and gas phase coexist. The typical material of the anode is Nickel-Yttria stabilized Zirconia cermet. The cathode must be porous in order to allow oxygen molecules to reach the electrode/electrolyte interface. The most commonly used cathode material is lanthanum manganite. It is often doped with strontium and referred to as lanthanum strontium manganite (LSM). The interconnect serves the purpose of connecting cells together to form a stack and it also acts as a collector of the electrical current. It functions as the electrical contact to the cathode while protecting it from the reducing atmosphere of the anode.

9.3 Chemical Equilibrium

For any isolated thermodynamic system undergoing a general process which does not have any particular restrictions, the sign of the change in entropy of the universe given by the second law of thermodynamics determines its spontaneity. However, many practical processes occur under the conditions of constant temperature and pressure. For reactions occurring at isothermal and isobaric conditions, the sign of the change in Gibbs function of the reaction gives the information about the spontaneity of the reaction.

Consider a hydrostatic system in mechanical and thermal equilibrium but not in chemical equilibrium. Suppose that the system is in contact with a reservoir at temperature $T$ and undergoes an infinitesimal irreversible process involving an exchange of heat $\delta Q$ from the reservoir. The process may involve a chemical reaction. Therefore, the entropy change of the universe is d$S_{0}+$d$S$, where the entropy change of the reservoir is d$S_{0}$ and the entropy change of the system is d$S$. During the infinitesimal irreversible process, the internal energy of the system changes by an amount d$U$, and an amount of work $p$d$V$ is performed by the system. Therefore, according to the first law,

$$\begin{aligned} \mathrm{d}Q=\mathrm{d}U+p\mathrm{d}V. \end{aligned}$$

(6.102)

According to the second law of thermodynamics, an irreversible process leads to the increase in the entropy of the universe. Thus, we may write d$S_{0}+$d$S>0$. Since d$S_{0}=-\delta Q/T$, we have

$$\begin{aligned} -\dfrac{\mathrm{d}Q}{T}+\mathrm{d}S>0\;\mathrm{or}\;\mathrm{d}Q-T\mathrm{d}S<0&\nonumber \\ \Rightarrow \mathrm{d}U+p\mathrm{d}V-T\mathrm{d}S<0&. \end{aligned}$$

(6.103)

By definition, the Gibbs function$G$ is given as $G=U+pV-TS$, which on differentiation gives $\mathrm{d}G=\mathrm{d}U+p\mathrm{d}V+V\mathrm{d}p-T\mathrm{d}S-S\mathrm{d}T$. When the conditions of constant temperature and pressure are imposed, the change in Gibbs function is given by

$$\begin{aligned} \mathrm{d}G=\mathrm{d}U+p\mathrm{d}V-T\mathrm{d}S \end{aligned}$$

(6.104)

Fig. 6.93

Variation of the Gibbs function of the reaction with the reaction advancement

From Eqs. 6.103 and 6.104, it is clear that the condition for the process to be irreversible and occur spontaneously is d$G<0$. Conversely, the non-spontaneity of an isothermal and isobaric process can be identified by the condition d$G>0$. Therefore, if d$G$ for a isothermal and isobaric chemical reaction system is negative, it means that the process will occur irreversibly until the equilibrium is reached at which, the Gibbs function $G$ will become minimum, i.e. d$G=0$ (refer to Fig. 6.93).

9.4 Bond Graph Model of Chemical Reaction Kinetics

Consider two reactants and product in the example reaction

$$\begin{aligned} \mathrm{2A + B }\overset{k_{+}}{\underset{k_{-}}{{\Huge \rightleftharpoons }}} \mathrm{2C,} \end{aligned}$$

(6.105)

where $k_{+}$ is the forward reaction rate coefficient and $k_{-}$ is the reverse reaction rate coefficient. The progress or advancement of a reaction can be represented in terms of mole numbers as $n_{i}\left( t\right) =n_{i}\left( 0\right) +\nu _{i}\xi \left( t\right)$, where $\xi \left( t\right)$ is the reaction advancement coordinate with $\xi \left( 0\right) =0$ and $\nu _{i}$ is the stoichiometric coefficient of the $i$th species. The stoichiometric coefficients of the species in this example reaction (refer to Eq. 6.105) are $\nu _{\mathrm{A}}=2$ , $\nu _{\mathrm{B}}=1$ and $\nu _{\mathrm{C}}=2$.

The time variation of the number of moles of the three species are related as d$n_{\mathrm{A}}=-\nu _{\mathrm{A}}\mathrm{d}\xi$, d$n_{\mathrm{B}}=-\nu _{\mathrm{B}}\mathrm{d}\xi$ and d$n_{\mathrm{C}}=\nu _{\mathrm{C}}$d$\xi$. As these quantities are perfect differentials, their integrations give the following:

$$\begin{aligned} n_{\mathrm{A}}(t)&=n_{\mathrm{A}}(0)-\nu _{\mathrm{A}}\xi \left( t\right) , \\ n_{\mathrm{B}}(t)&=n_{\mathrm{B}}(0)-\nu _{\mathrm{B}}\xi \left( t\right) , \\ n_{\mathrm{C}}(t)&=n_{\mathrm{C}}(0)+\nu _{\mathrm{C}}\xi \left( t\right) , \end{aligned}$$

where $\xi \left( 0\right) =0$.

The reaction advancement coordinate may be thought of as a generalized displacement whose time derivative defines the reaction rate [35]. The corresponding effort, which drives the reaction, is called affinity. The Gibbs free energy is the maximum amount of non-expansion work that can be extracted from a closed system through a completely reversible process. The change in Gibbs free energy of a system which is maintained at constant temperature and pressure is given as

$$\begin{aligned} \mathrm{d}G=\left( \sum \limits _{i\in \mathrm{P}}\mu _{i}\nu _{i}-\sum \limits _{i\in \mathrm{R}}\mu _{i}\nu _{i}\right) \mathrm{d}\xi , \end{aligned}$$

(6.106)

where subscript $i$ is used to represent the sums over the product and the reactant components and $\mu$ is the chemical potential. In this example, P= $\left[ \begin{array}{cc} \mathrm{A}&\mathrm{B} \end{array} \right]$ and R=$\left[ \mathrm{C}\right]$.

The affinity of the reaction is given by

$$\begin{aligned} A=\sum \limits _{i\in \mathrm{R}}\mu _{i}\nu _{i}-\sum \limits _{i\in \mathrm{P}}\mu _{i}\nu _{i}, \end{aligned}$$

(6.107)

where the quantity $A_{\mathrm{F}}=\sum \limits _{i\in \mathrm{R}}\mu _{i}\nu _{i}$ is defined as the forward affinity and the quantity $A_{ \mathrm{R}}=\sum \limits _{i\in \mathrm{P}}\mu _{i}\nu _{i}$ is defined as the reverse affinity. Therefore, Eq. 6.106 can be written as

$$\begin{aligned} \dfrac{\mathrm{d}G}{\mathrm{d}\xi }=-A. \end{aligned}$$

(6.108)

The affinity may be defined as a generalized effort variable in the bond graph model. At the equilibrium, the affinity of the reaction is zero, i.e., d$G/\mathrm{d}\xi =0$ or $A=0$ and $A_{ \mathrm{F}}=A_{\mathrm{R}}$.

The reaction rates are defined by the law of mass action. The law of mass action states that the rate of an elementary reaction (a reaction that proceeds through only one mechanistic step) is proportional to the product of the concentrations of the participating molecules. Therefore, the net rate of the reversible chemical reaction considered in Eq. 6.105 is given as

$$\begin{aligned} \dot{\xi }=k_{+}c_{\mathrm{A}}^{\mathrm{2}}c_{\mathrm{B}}-k_{-}c_{\mathrm{C}}^{\mathrm{2 }}, \end{aligned}$$

(6.109)

where $c$ refers to the concentration of the species ‘$i$’. The concentration of a specific component is defined as the ratio of the number of moles of that component to the total number of moles of products and reactants.

The true bond graph model for the kinetics of the reaction (refer to Eq. 6.105) is shown in Fig. 6.94. The effort and flow variables in the model are chemical potential and mole flow rate, respectively. The chemical potential of the component ‘$i$’ is given as

$$\begin{aligned} \mu _{i}=\mu _{i \mathrm{,0}}\left( T,p\right) +\mathrm{R}T\ln \left( \dfrac{c_{i}}{c_{\Sigma }} \right) , \end{aligned}$$

(6.110)

where $c_{i}$ is the concentration of the component ‘$i$’, $c_{\Sigma }$ is the reference concentration (usually taken as unity) and $\mu _{i\mathrm{,0}}$ is the standard chemical potential. Equation 6.110 can be written in terms of number of moles ‘$n$’ as

$$\begin{aligned} \mu _{i}=\mu _{i\mathrm{,0}}\left( T,p\right) +\mathrm{R}T\ln \left( \dfrac{n_{i} }{n_{\Sigma }}\right) , \end{aligned}$$

(6.111)

which forms the constitutive relation for the C-elements of the bond graph model.

From Eq. 6.110, the concentrations of the reacting species can be represented in terms of their chemical potentials as

$$\begin{aligned} c_{i}=\exp \left( \dfrac{\mu _{i}-\mu _{i\mathrm{,0}}\left( T,p\right) }{\mathrm{R}T}\right) . \end{aligned}$$

(6.112)

The effort in the bond number 7 shown in Fig. 6.94 denotes the forward affinity of the reaction and is given by $e_{7}=A_{ \mathrm{F}}=2\mu _{\mathrm{A}}+\mu _{\mathrm{B}}$. Similarly, the effort in bond number 8 denotes the reverse affinity of the reaction and is given by $e_{8}=A_{ \mathrm{R}}=2\mu _{\mathrm{C}}$.

Fig. 6.94

The bond graph model of an isothermal and isobaric chemical reaction

The non-equilibrium part of the chemical reaction kinetics is represented as an R-field whose inputs are the forward and reverse affinities and the output is the reaction rate. The constitutive relation of the R-field is formulated using the mass action law. From Eqs. 6.109 and 6.112, the flows in the bonds numbered 7 and 8 are given as

$$\begin{aligned} f_{7}=f_{8}&= \dot{\xi }=k_{+}\exp \left( \dfrac{2\mu _{\mathrm{A}}-2\mu _{\mathrm{A,0}}}{\mathrm{R}T}\right) \exp \left( \dfrac{\mu _{\mathrm{B}}-\mu _{\mathrm{B,0}}}{\mathrm{R}T} \right) -k_{-}\exp \left( \dfrac{2\mu _{\mathrm{C}}-2\mu _{\mathrm{C,0}}}{\mathrm{R }T}\right)\nonumber \\&=k_{+}\exp \left( \dfrac{-2\mu _{\mathrm{A,0}}-\mu _{\mathrm{B,0}}}{\mathrm{R}T} \right) \exp \left( \dfrac{A_{\mathrm{F}}}{\mathrm{R}T}\right) -k_{-}\exp \left( \dfrac{-2\mu _{\mathrm{C,0}}}{\mathrm{R}T}\right) \exp \left( \dfrac{A_{\mathrm{R}} }{\mathrm{R}T}\right) , \end{aligned}$$

where, $\dot{\xi }$ has the unit of mol s$^{-1}$. This resistance field satisfies the Onsager reciprocity requirements [54]. Moreover, the network representation [61] of the chemical reactions helps in integrating the kinetic (rate laws defined by Eq. 6.109 ) and thermodynamic (affinities as defined in Eq. 6.107 as driving forces) points of view.

9.5 SOFC Modeling

Different levels of modeling of the SOFC system are possible. One has to consider spatial distribution of pressure, temperature and concentration. Such models are complex to solve and usually need a combined thermal and computational fluid dynamics (CFD) model. A full model involves slow (hydraulic and thermal) and fast (reaction, diffusion, $\ldots$) phenomena and therefore, multi-scale models are needed [23]. Fast dynamics is generally related to microscopic level. Let us consider a so-called zero-dimensional macroscopic model of the SOFC system. Such a model is good enough to develop the SOFC control system.

The assumptions for the model are

• The water formed due to the reaction is in the vapor form. All the gases involved are assumed to be ideal. This assumption is valid because of the low pressure and high operating temperatures.

• The fuel considered in this model is pure hydrogen. The oxidant is air with oxygen and nitrogen as its primary constituents.

• As the cell is well insulated, the heat loss to the surrounding is neglected.

• As the fast dynamics are irrelevant from the control perspective, the diffusion process is modeled through an approximation.

9.5.1 Storage of a Two-Species Gas Mixture

The SOFC channels on the anode side contain the hydrogen and water vapor, while the cathode side has nitrogen and oxygen. On the anode side hydrogen is consumed in the electrochemical reaction and the water vapor is produced, while on the cathode side oxygen is consumed. The nitrogen on the cathode side does not participate in the reaction. A storage element capable of representing the energy storage in a two-species gas mixture is necessary for modeling this scenario.

Consider that a mixture of two gases is contained in a collapsible chamber, which allows heat transfer from and to the surroundings. It is assumed that individual gases can independently flow either into or out of the chamber. Allowing the individual gas mass flow rates in proportion to their mass fractions in the mixture can also incorporate the mass flow of the mixture as a whole. The change of internal energy of the two gases in the mixture in terms of time derivatives is given by

$$\begin{aligned} \dot{U}=\dfrac{\partial U}{\partial V}\dot{V}+\dfrac{\partial U}{\partial S} \dot{S}+\dfrac{\partial U}{\partial m_{1}}\dot{m}_{1}+\dfrac{\partial U}{ \partial m_{2}}\dot{m}_{2}, \end{aligned}$$

(6.113)

where subscripts 1 and 2 identify the species, $U$, $V$, and $S$ are the total internal energy, common volume occupied by the species, and total entropy, respectively, $m$ is the mass and superposed (.) indicates time derivative.

From the well-known thermodynamic relations, $\partial U/\partial V=-p$, $\partial U/\partial S=T$, $\partial U/\partial m_{1}=\mu _{1}$ and $\partial U/\partial m_{2}=\mu _{2}$, it is evident that the internal energy of the volume of the gases change due to four distinct power exchanges which can be represented by the products of the corresponding effort and flow variables. Therefore, the energy storage in the gas mixture can be represented as a four-port C-field as shown in Fig. 6.95. This C-field (an extension of the model in [11]) has four power ports: the flow and effort variables for the mechanical port are $\dot{V}$ and $p$, respectively; those for the thermal port are $\dot{S}$ and $T$, respectively; and those for the material ports are $\dot{m}^{\prime }$s and $\mu$’s, respectively.

Fig. 6.95

Two species of gases represented in a C-field

We start from the ideal gas equation of state ($pv=RT$, $v$ being the specific volume and $R$ being characteristic gas constant) and the definition of specific heat capacity at constant volume ( $\mathrm{d}u=c_{ \mathrm{v}}\mathrm{d}T$, $c_{\mathrm{v}}$ being the specific heat at constant volume). According to the fundamental thermodynamic relation [77], the change in the specific entropy ($s$) of an ideal gas (gas species #1) in terms of the specific internal energy ($u$), the specific volume ($v$), the partial pressure ($p_{i}$, $i=1,2$) and the equilibrium temperature ( $T)$ is given by

$$\begin{aligned} \mathrm{d}s_{1}=\dfrac{\mathrm{d}u_{1}}{T}+\dfrac{p_{1}\mathrm{d}v_{1}}{T}=\dfrac{ c_{\mathrm{v1}}\mathrm{d}T}{T}+\dfrac{R_{1}\mathrm{d}v_{1}}{v_{1}}. \end{aligned}$$

(6.114)

Integrating Eq. 6.114 from an initial state (indicated by superscript 0) to a final state with the assumption of constant specific heat capacities and then writing the specific quantities in terms of the absolute quantities gives

$$\begin{aligned} s_{1}=\dfrac{S_{1}^{0}}{m_{1}^{0}}+\ln \left\{ \left( \dfrac{T}{T^{0}} \right) ^{c_{\mathrm{v1}}}\left( \dfrac{Vm_{1}^{0}}{V^{0}m_{1}}\right) ^{R_{1}}\right\} . \end{aligned}$$

(6.115)

Similarly, for gas species #2, which occupies the same volume and is at same temperature, we obtain

$$\begin{aligned} s_{2}=\dfrac{S_{2}^{0}}{m_{2}^{0}}+\ln \left\{ \left( \dfrac{T}{T^{0}} \right) ^{c_{\mathrm{v2}}}\left( \dfrac{Vm_{2}^{0}}{V^{0}m_{2}}\right) ^{R_{ {2}}}\right\} . \end{aligned}$$

(6.116)

Multiplying Eq. 6.115 with $m_{1}$ and Eq. 6.116 with $m_{2}$, we get the following expressions for the entropies of the gas species #1 and #2.

$$\begin{aligned} S_{1}-\dfrac{S_{1}^{0}m_{1}}{m_{1}^{0}}&=\ln \left\{ \left( \dfrac{T}{T^{0} }\right) ^{m_{1}c_{\mathrm{v1}}}\left( \dfrac{Vm_{1}^{0}}{V^{0}m_{1}}\right) ^{m_{1}R_{1}}\right\} , \nonumber \\ S_{2}-\dfrac{S_{2}^{0}m_{2}}{m_{2}^{0}}&=\ln \left\{ \left( \dfrac{T}{T^{0} }\right) ^{m_{2}c_{\mathrm{v2}}}\left( \dfrac{Vm_{2}^{0}}{V^{0}m_{2}}\right) ^{m_{2}R_{2}}\right\} . \end{aligned}$$

(6.117)

The total entropy of a mixture of gases is given by the sum of the entropies of the individual gases: $S=\left( S_{1}+S_{2}\right)$. Then, the temperature of the gases can be written as a function of the four state variables ( $m_{1}, m_{2}, V\;\mathrm{and}\;S$):

$$\begin{aligned} {\small T=T}^{0}{\small e}^{\alpha }\left( \dfrac{{\small V}}{{\small V}^{0}} \right) ^{-\left( \dfrac{{\tiny m}_{1}{\tiny R}_{1}{\tiny +m}_{2}{\tiny R} _{2}}{{\tiny m}_{1}{\tiny c}_{\mathrm{v1}}{\tiny +m}_{2}{\tiny c}_{\mathrm{v2}}} \right) } \nonumber \\ \times \left( \dfrac{{\small m}_{1}}{ {\small m}_{1}^{0}}\right) ^{\left( \dfrac{{\tiny m}_{1}{\tiny R}_{1}}{ {\tiny m}_{1}{\tiny c}_{\mathrm{v1}}{\tiny +m}_{2}{\tiny c}_{\mathrm{v2}}} \right) }\left( \dfrac{{\small m}_{2}}{{\small m}_{2}^{0}}\right) ^{\left( \dfrac{{\tiny m}_{2}{\tiny R}_{2}}{{\tiny m}_{1}{\tiny c}_{\mathrm{v1}}{\tiny +m}_{2}{\tiny c}_{\mathrm{v2}}}\right) }, \end{aligned}$$

(6.118)

where $\alpha =\dfrac{{\small S}}{{\small m}_{1}c_{\mathrm{v1}}{\small +m}_{2} {\small c}_{\mathrm{v2}}}$ ${\small -}$ $\dfrac{{\small m}_{1}{\small S} _{1}^{0}}{{\small m}_{1}{\small m}_{1}^{0}{\small c}_{\mathrm{v1}}{\small +m} _{1}^{0}{\small m}_{2}{\small c}_{\mathrm{v2}}}$ ${\small -}$ $\dfrac{ {\small m}_{2}{\small S}_{2}^{0}}{{\small m}_{1}{\small m}_{2}^{0}{\small c} _{\mathrm{v1}}{\small +m}_{2}^{0}{\small m}_{2}{\small c}_{\mathrm{v2}}}.$

The internal energy of the gas mixture is obtained as the sum of the internal energies of both the gases in the chamber, i.e., $U=m_{1}c_{\mathrm{v1} }T+m_{2}c_{\mathrm{v2}}T$. The total pressure in the chamber is then obtained by summing the partial pressures, i.e., $p=m_{1}R_{1}T/V+m_{2}R_{2}T/V$, where $T$ is given by Eq. 6.118. The same result can also be obtained by taking the partial derivative of the total internal energy with respect to the total volume.

$$\begin{aligned} {\small p=T}^{0}\left( \dfrac{{\small m}_{1}{\small R}_{1}{\small +m}_{2} {\small R}_{2}}{{\small V}}\right) {\small e}^{\alpha }\left( \dfrac{{\small V}}{{\small V}^{0}}\right) ^{-\left( \dfrac{{\tiny m}_{1}{\tiny R}_{1}{\tiny +m}_{2}{\tiny R}_{2}}{{\tiny m}_{1}{\tiny c}_{\mathrm{v1}}{\tiny +m}_{2}{\tiny c}_{\mathrm{v2}}}\right) } \nonumber \\ \times \left( \dfrac{{\small m}_{1}}{ {\small m}_{1}^{0}}\right) ^{\left( \dfrac{{\tiny m}_{1}{\tiny R}_{1}}{ {\tiny m}_{1}{\tiny c}_{\mathrm{v1}}{\tiny +m}_{2}{\tiny c}_{\mathrm{v2}}} \right) }\left( \dfrac{{\small m}_{2}}{{\small m}_{2}^{0}}\right) ^{\left( \dfrac{{\tiny m}_{2}{\tiny R}_{2}}{{\tiny m}_{1}{\tiny c}_{\mathrm{v1}}{\tiny +m}_{2}{\tiny c}_{\mathrm{v2}}}\right) }. \end{aligned}$$

(6.119)

Likewise, the chemical potentials of the gases can be obtained by taking the partial derivative of $U$ with respect to their corresponding masses. Alternatively, the chemical potential of gas #1 can be given as $\mu _{1}=u_{1}+p_{1}v_{1}-Ts_{1}=c_{ \mathrm{v1}}T+R_{1}T-Ts_{1}=h_{1}-Ts_{1}.$ Noting that the specific enthalpy $h_{1}=h_{1}^{0}+\int c_{ \mathrm{p}}$d$T$ and $s_{1}=s_{1}^{0}+\int \left( c_{\mathrm{p}}/T\right)$d$T-R_{1}\ln \left( p_{1}/p_{1}^{0}\right)$, we get

$$\begin{aligned} \mu _{1}=\mu _{1}^{0}\left( T\right) +R_{1}T\ln \left( \dfrac{p_{1}}{ p_{1}^{0}}\right) , \end{aligned}$$

(6.120)

where $\mu _{1}^{0}\left( T\right)$ is purely a function of the temperature. The partial pressure of the gas species #1 and the temperature of the mixture in Eq. 6.120 are written in terms of the state variables using the earlier expressions (refer to Eqs. 6.118 and 6.119). The chemical potential of gas #2 is obtained in a similar fashion as $\mu _{2}=\mu _{2}^{0}\left( T\right) +R_{2}T\ln \left( p_{2}/p_{2}^{0}\right) .$ Equations 6.118 $-$ 6.120 are the constitutive relations of the four-port C-field as they give the effort variables ($\mu _{1}, \ \mu _{2}, \ p\mathrm{\ and\ }T$) in terms of the four state variables ($m_{1}, \ m_{2}, \ V \mathrm{\ and\ }S$), which are obtained by integrating the flow variables in the bonds of the four-port C-field shown in Fig. 6.95.

9.5.2 Bond Graph Model for Convection of a Gas Mixture

An R-field represents the convection of a two gas mixture [22]. The most important element in the expanded model of the MR-element is the RS-field element (see Fig. 6.96). This element receives the downstream side temperature and the information of the valve position ($x$), the upstream side chemical potentials and temperature, and the downstream side chemical potentials to calculate the mass and entropy flow rates. Note that all these variables are inputs to the MR-element. To maintain the clarity of the figure, the connections needed to explicitly show these modulations are not drawn. In Fig. 6.96, subscripts u and d refer to upstream and downstream sides, respectively.

Fig. 6.96

Bond graph representation for convection of a two-component gas mixture

The submodel in Fig. 6.96 receives information of six effort variables and computes six flow variables without the use of integration and/or differentiation. Therefore, this submodel can be represented as an encapsulated R-field (a six-port element MR in Fig. 6.96). From the continuity equation, the mass flow rate of a particular gas is the same for the inlet and the outlet side. This reduces the total number of independent flow variables to four (see Fig. 6.96). Then the constitutive relation of the nonlinear resistive field element is given as

$$\begin{aligned} \left\{ \dot{S}_{\mathrm{u}},\dot{S}_{\mathrm{d}},\dot{m}_{1},\dot{m}_{2}\right\} ^{ \mathrm{T}}=\varPhi _{\mathrm{R}}\left\{ \left( T_{\mathrm{u}},T_{\mathrm{d}},\mu _{ \mathrm{1u}},\mu _{\mathrm{1d}},\mu _{\mathrm{2u}},\mu _{\mathrm{2d}}\right) ^{\mathrm{T}}\right\} . \end{aligned}$$

(6.121)

where, $\varPhi _{\mathrm{R}}(.)$ is a vector-valued function.

The overall mass flow rate ($\dot{m}$) of the mixture is imposed at the $1_{ \dot{m}}$ junction by the modulated RS-field element in Fig. 6.96 and it is given by the linear nozzle equation:

$$\begin{aligned} \dot{m}=K\left(p_{\mathrm{u}}-p_{\mathrm{d}}\right) . \end{aligned}$$

(6.122)

Although the total upstream and downstream side pressures are needed to calculate mass flow rate, they can indeed be calculated from the chemical potentials and temperatures. The individual mass flow rates of the two gases are then realized through the MTF elements shown in Fig. 6.96 as $\dot{m}_{1}=\dot{m}w_{\mathrm{1u}}$ and $\dot{m}_{2}=\dot{ m}w_{\mathrm{2u}}$. The upstream mass fractions $w_{\mathrm{1u}}$ and $w_{\mathrm{2u}}$ are obtained from the upstream side storage element (the C-field in Fig. 6.95), i.e., $w_{\mathrm{1u}}=m_{\mathrm{1u}}/\left(m_{\mathrm{1u}}+m_{\mathrm{2u}}\right)$, $w_{\mathrm{2u}}=m_{\mathrm{2u}}/\left(m_{\mathrm{1u} }+m_{\mathrm{2u}}\right)$ and $w_{\mathrm{1u}}+w_{\mathrm{2u}}=1$, where $m_{ \mathrm{1u}}$ and $m_{\mathrm{2u}}$ are the contemporary masses (state variables) in the upstream side control volume.

The entropy flow rate associate with the mass flow rate is calculated by means of a transformer element (between junctions $1_{\dot{S}}$ and $1_{\dot{ m}}$), which is modulated by the specific entropy of the upstream side gases. This information of the upstream side specific entropy can either be obtained directly from the upstream side storage element or if a standalone scheme is required, it can be calculated from the upstream side $\mu$’s and $T$’s (which are inputs of the MR-element) as

$$\begin{aligned} s_{\mathrm{u}}=c_{\mathrm{p1}}w_{\mathrm{1u}}+c_{\mathrm{p2}}w_{\mathrm{2u}}-\dfrac{ \left( \mu _{\mathrm{1u}}w_{\mathrm{1u}}+\mu _{\mathrm{2u}}w_{\mathrm{2u}}\right) }{ T_{\mathrm{u}}}, \end{aligned}$$

(6.123)

where $c_{\mathrm{p}}$ is the specific heat at constant pressure.

The entropy flow rate from the upstream side is given as $\dot{S}_{\mathrm{u}}= \dot{m}s_{\mathrm{u}}$. The R-field represents the change in the intensive variables between the upstream and the downstream sides. The temperatures, pressures, and the chemical potentials of the gas mixture in the upstream and the downstream sides are imposed by the storage elements on the corresponding sides. Due to this, there is an enthalpy difference between the upstream and downstream sides, which can be represented as the relation between the changes in the intensive variables by using the Gibbs-Duhem equation [16] as

$$\begin{aligned} v \ \left(p_{\mathrm{u}}-p_{\mathrm{d}}\right) =s_{\mathrm{u}}\left(T_{ \mathrm{u}}-T_{\mathrm{d}}\right) +w_{\mathrm{1u}}\left( \mu _{\mathrm{1u}}-\mu _{ \mathrm{1d}}\right) +w_{\mathrm{2u}}\left( \mu _{\mathrm{2u}}-\mu _{\mathrm{2d} }\right) . \end{aligned}$$

(6.124)

This relation is enforced by the $1_{\dot{m}}$-junction in Fig. 6.96. Due to the enthalpy difference between the upstream and downstream side gases, entropy is generated in the resistive field. Using the principle of power conservation, the irreversible entropy generated $\dot{S}_{\mathrm{gen}}$ can be given as

$$\begin{aligned} \dot{S}_{\mathrm{gen}}=\dfrac{\dot{m}v\left( P_{\mathrm{u}}-P_{\mathrm{d}}\right) }{T_{\mathrm{d}}}=\dfrac{\dot{m}\left( s_{\mathrm{u}}\left( T_{\mathrm{u}}-T_{ \mathrm{d}}\right) +w_{\mathrm{1u}}\left( \mu _{\mathrm{1u}}-\mu _{\mathrm{1d} }\right) +w_{\mathrm{2u}}\left( \mu _{\mathrm{2u}}-\mu _{\mathrm{2d}}\right) \right) }{T_{\mathrm{d}}},\nonumber \\ \end{aligned}$$

(6.125)

where $s_{\mathrm{u}}\left( T_{\mathrm{u}}-T_{\mathrm{d}}\right) +w_{\mathrm{1u} }\left( \mu _{\mathrm{1u}}-\mu _{\mathrm{1d}}\right) +w_{\mathrm{2u}}\left( \mu _{ \mathrm{2u}}-\mu _{\mathrm{2d}}\right)$ and $T_{\mathrm{d}}$ are effort inputs to the RS-element and $\dot{m}$ is calculated internally from the constitutive relation of the RS-element (see Eq. 6.122). The downstream side entropy flow rate is the sum of the upstream side entropy flow rate ($\dot{S} _{\mathrm{u}}$ imposed at $1_{\dot{S}}$-junction by the MTF-element) and the irreversible entropy generated (Eq. 6.125). This sum is realized by means of the zero-junction shown in Fig. 6.96.

The upstream and downstream pressures, which are needed in Eq. 6.122, can either be read directly from the upstream and downstream side storage elements ($C$-fields) or can be calculated as functions of $\mu$’s and $T$’s (the input variables to the MR-element) as

$$\begin{aligned} p_{\mathrm{1u}}&=\dfrac{R_{1}T_{\mathrm{u}}}{v_{\mathrm{u1}}}=p_{\mathrm{u1}}^{\mathrm{0 }}\exp \left( \dfrac{\mu _{\mathrm{1u}}}{T_{\mathrm{u}}R_{1}}-\dfrac{\mu _{\mathrm{1u}}^{\mathrm{0}}}{T_{\mathrm{u}}^{\mathrm{0}}R_{1}}\right) \left( \dfrac{T_{\mathrm{u}}}{T_{\mathrm{u}}^{\mathrm{0}}}\right) ^{\dfrac{c_{\mathrm{v1}}}{R_{1}}}, \nonumber \\ p_{\mathrm{2u}}&=\dfrac{R_{2}T_{\mathrm{u}}}{v_{\mathrm{u2}}}=p_{\mathrm{u2}}^{\mathrm{0 }}\exp \left( \dfrac{\mu _{\mathrm{2u}}}{T_{\mathrm{u}}R_{2}}-\dfrac{\mu _{\mathrm{2u}}^{\mathrm{0}}}{T_{\mathrm{u}}^{\mathrm{0}}R_{2}}\right) \left( \dfrac{T_{\mathrm{u}}}{T_{\mathrm{u}}^{\mathrm{0}}}\right) ^{\dfrac{c_{\mathrm{v2}}}{R_{2}}}. \end{aligned}$$

(6.126)

The total upstream side pressure is $p_{\mathrm{u}}=p_{\mathrm{1u}}+p_{\mathrm{2u} }$. The total downstream side pressure can also be expressed similarly.

9.5.3 True Bond Graph Model of the SOFC

The true bond graph model of the SOFC is shown in Fig. 6.97. It uses the four-port C-field for representing the energy storage of the gases inside the anode and cathode flow channels. It also uses the R-field representation discussed in the last section for modeling the convection at the inlet and the outlet of the SOFC channels. The subscripts labeling various elements and power variables are as follows: an (anode), ca (cathode), H (hydrogen), O (Oxygen), N (nitrogen), W (water), AS (air source), HS (hydrogen source), M (Membrane electrode assembly or MEA), ai (anode inlet), ci (cathode inlet), ao (anode outlet), co (cathode outlet), I1 (Interconnect on anode side), I2 (Interconnect on cathode side), PL (polarisation losses), r (reaction), and ENV (environment).

As the volumes of both the channels remain constant, the mechanical ports of the C-fields ($p$-$\dot{V}$ port of Fig. 6.95) are not shown in Fig. 6.97. The mass and entropy balances of the anode and cathode channel control volumes are given by the corresponding zero junctions in Fig. 6.97. The $0_{T_{\mathrm{an}}}$ and the $0_{T_{\mathrm{ca} }}$ junctions give the entropy balances for the anode channel and the cathode channel control volumes, respectively. The $0_{\mu _{\mathrm{H}}}$, $0_{\mu _{\mathrm{W}}}$, $0_{\mu _{\mathrm{O}}}$ and $0_{\mu _{\mathrm{N}}}$ junctions give the mass balances for the hydrogen, water vapor, oxygen, and nitrogen gases, respectively, in the control volumes. The $0_{T_{\mathrm{M}}}$ junction gives the entropy balance at the MEA solid control volume.

The capacitive elements and fields in the model represent equilibrium thermodynamics part of the model. As the simulation precedes, the matter inside the control volume represented by these elements changes reversibly from one equilibrium state to the next, i.e., the process is assumed to be quasi-static. The R-fields represent the non-equilibrium parts of the model and they introduce the irreversibilities into the system. The MR-elements introduce the irreversibility due to mass convection into the system. The R-field element represented by ‘RS’ in Fig. 6.97 introduces the irreversibility due to the over-voltage phenomena (ohmic, concentration and activation losses). The other R-field elements introduce the irreversibilities due to the heat transfer phenomena.

Fig. 6.97

True bond graph model of the SOFC

The inlet and outlet valve resistances are modeled by the MR-fields described in Fig. 6.96, where subscripts are used to identify them. The valve resistances in the MR-fields may be controlled by modifying the variables for the stem positions. Note that although only hydrogen gas flows through the anode side inlet valve, the information of chemical potential of water vapor ($\mu _{\mathrm{W}}$) inside the anode channel is required for computing the downstream side pressure, which is supplied by an information bond in Fig. 6.97. Similarly, the additional information of the chemical potentials of nitrogen and oxygen in the atmosphere are required in the anode channel outlet valve model to calculate the downstream side pressure, which is provided by the source of efforts as shown in Fig. 6.97. The downstream side entropy flow is the sum of the upstream side entropy flow and the entropy generated due to the enthalpy difference between the upstream and downstream sides (Eq. 6.125).

In this model, the chemical potentials of the gases not only drive the electrochemical reaction but also, along with temperatures, determine the flow of the gases in and out of the channels. The transformation of power from the chemical domain into the electrical domain is implemented by the $1_{\dot{\xi }}$ junction and the transformers surrounding it as shown in Fig. 6.97.

The change in the Gibbs free energy of the system is given as

$$\begin{aligned} \mathrm{d}G=V\mathrm{d}p-S\mathrm{d}T+\mu \mathrm{d}m. \end{aligned}$$

(6.127)

Assuming constant temperature and pressure, the change in the Gibbs free energy of the reaction is obtained as

$$\begin{aligned} \mathrm{d}G=\dfrac{\partial G}{\partial n_{\mathrm{W}}}\mathrm{d}n_{\mathrm{W}}- \dfrac{\partial G}{\partial n_{\mathrm{H}}}\mathrm{d}n_{\mathrm{H}}-\dfrac{ \partial G}{\partial n_{\mathrm{O}}}\mathrm{d}n_{\mathrm{O}}. \end{aligned}$$

(6.128)

Note that the temperature and the pressure of the system may change during the system’s dynamics. However, Eq. 6.128 is assumed to be valid for each instantaneous values of pressure and temperature of the system. The reaction coordinate ($\xi$) is defined such that d$n_{\mathrm{H} }=-\nu _{\mathrm{H}}\mathrm{d}\xi$, d$n_{\mathrm{O}}=-\nu _{\mathrm{O}}\mathrm{d}\xi$ and d$n_{ \mathrm{W}}=\nu _{\mathrm{W}}$d$\xi$. Using these relations and the definition of the chemical potential, Eq. 6.128 can be written as

$$\begin{aligned} \Delta G=\left( \mu _{\mathrm{W}}\nu _{\mathrm{W}}-\mu _{\mathrm{H}}\nu _{\mathrm{H} }-\mu _{\mathrm{O}}\nu _{\mathrm{O}}\right) \Delta \xi . \end{aligned}$$

(6.129)

If unit mole of fuel (hydrogen) is considered then $\Delta \xi =1$. Therefore, the change in the Gibbs free energy per mole of fuel is given by

$$\begin{aligned} \Delta G=\mu _{\mathrm{W}}\nu _{\mathrm{W}}-\mu _{\mathrm{H}}\nu _{\mathrm{H}}-\mu _{ \mathrm{O}}\nu _{\mathrm{O}}. \end{aligned}$$

(6.130)

Note that the chemical potentials are in J.mol$^{-1}$ in Eq. 6.130. Under reversible conditions, this change in the Gibbs free energy is converted entirely into electrical energy. Therefore, from the energy balance, the reversible cell voltage can be obtained as

$$\begin{aligned} V_{\mathrm{r}}=-\dfrac{\Delta G}{n_{\mathrm{e}}F}, \end{aligned}$$

(6.131)

where the denominator gives the charge of the total number of electrons participating in the reaction per mole of the fuel and F is the Faraday’s number. Equation 6.131 can further be written in terms of the partial pressures of the reactant and the product gases and is called the Nernst equation. The Nernst equation is used to calculate the effect of the change in the partial pressures of the reacting species on the reversible cell voltage. Note that the minus sign in Eq. is required to obtain a positive value of voltage because the change in the Gibbs free energy per mol as defined in Eq. 6.130 is negative (as the free energy of the products is less than the free energy of the reactants).

The chemical potentials are calculated in J.kg$^{-1}$ in the anode and cathode channel C-fields of the model. The three transformers shown in the effort activated bonds around the $1_{ \dot{\xi }}$ junction have factors of ‘1,000/$M_{i}$’ in order to convert the chemical potentials into J.mol$^{-1}$. The $1_{\dot{\xi }}$ junction shown in Fig. 6.97 enforces the following relationship, which defines the negative of the change in Gibbs free energy per mol of fuel for the reaction.

$$\begin{aligned} -\Delta G=\dfrac{\nu _{\mathrm{H}}M_{\mathrm{H}}\mu _{\mathrm{H}}+\nu _{\mathrm{O} }M_{\mathrm{O}}\mu _{\mathrm{O}}-\nu _{\mathrm{W}}M_{\mathrm{W}}\mu _{\mathrm{W}}}{1{,}000} \end{aligned}$$

(6.132)

The reversible cell voltage, which is defined by the Nernst equation, is realised by means of a transformer element (with modulus $n_{\mathrm{e}}F$) in Fig. 6.97. When the reaction system is in equilibrium, the change in the molar Gibbs free energy ($\Delta G$) is zero. Therefore, the reversible voltage as predicted by the Nernst equation is also zero. When the reaction system is forced out of equilibrium (i.e., when the concentrations of the reactants and the products differ from the equilibrium concentrations), the reversible open circuit voltage ($V_{r}$) can be calculated by using the Nernst equation. However, the reaction cannot proceed as the circuit is not closed. But once the circuit is closed (as we try to draw current from the cell), the irreversibilities come into play and result in voltage losses.

The mole flow rate of the reaction ($\dot{\xi }$), which can be considered as the reaction rate, is related to the mole flow rates of consumption and production of the reactants and products, respectively, as

$$\begin{aligned} \dot{\xi }=\dfrac{\dot{n}_{\mathrm{W}}^{\mathrm{r}}}{\nu _{\mathrm{W}}}=-\dfrac{ \dot{n}_{\mathrm{H}}^{\mathrm{r}}}{\nu _{\mathrm{H}}}=-\dfrac{\dot{n}_{\mathrm{O}}^{ \mathrm{r}}}{\nu _{\mathrm{O}}}. \end{aligned}$$

(6.133)

The reaction mole flow rate and the current ($i$) are related as

$$\begin{aligned} i=\dot{\xi }n_{\mathrm{e}}F. \end{aligned}$$

(6.134)

Therefore, the relations between the mass-flow rates (in kg.s$^{-1}$) of hydrogen, oxygen, and water vapor taking part in the reaction and the current drawn by the load are given as

$$\begin{aligned} i=\dfrac{1{,}000n_{\mathrm{e}}F\dot{m}_{\mathrm{W}}^{\mathrm{r}}}{\nu _{\mathrm{W}}M_{ \mathrm{W}}}=-\dfrac{1{,}000n_{\mathrm{e}}F\dot{m}_{\mathrm{H}}^{\mathrm{r}}}{\nu _{ \mathrm{H}}M_{\mathrm{H}}}=-\dfrac{1{,}000n_{\mathrm{e}}F\dot{m}_{\mathrm{O}}^{\mathrm{r}} }{\nu _{\mathrm{O}}M_{\mathrm{O}}} \end{aligned}$$

(6.135)

and they are realized through the $1_{\dot{\xi }}$ junction and the set of transformers in the bonds surrounding it as shown in Fig. 6.97. The current, $i$, drawn by an un-modeled external load is represented by a source of flow.

The theoretical open-circuit voltage ($V_{\mathrm{r}}$) is the maximum voltage that can be achieved by a fuel cell under specific operating conditions. However, the voltage of an operating cell, which is equal to the voltage difference between the cathode and the anode, is generally lower than this. As current is drawn from a fuel cell, the cell voltage falls due to the internal resistances and overvoltage losses. The electrode overvoltage ($\eta$) losses are associated with the electrochemical reactions taking place at the electrode/electrolyte interfaces and can be divided into concentration $\left( \eta _{\mathrm{act}}\right)$ and activation $\left( \eta _{\mathrm{conc}}\right)$ over-voltages. The actual cell voltage is generally obtained by subtracting all the voltage losses from the open circuit voltage.

Three different kinds of voltage losses or overvoltages contribute to the cell irreversibility. Activation overvoltage refers to the overpotential required to exceed the activation energy barrier so that the electrode reactions proceed at the desired rate. The anodic and the cathodic activation overvoltages are governed by the Butler-Volmer equation [9], which for a transfer coefficient value of 0.5 are given as

$$\begin{aligned} \eta _{\mathrm{act,an}}&=\dfrac{2RT_{\mathrm{M}}}{n_{\mathrm{e}}F}\sinh ^{-1}\left( \dfrac{0.5i}{i_{\mathrm{0,an}}}\right) \end{aligned}$$

(6.136)

$$\begin{aligned} \mathrm{and}\eta _{\mathrm{act,ca}}&=\dfrac{2RT_{\mathrm{M}}}{n_{ \mathrm{e}}F}\sinh ^{-1}\left( \dfrac{0.5i}{i_{\mathrm{0,c}}}\right) . \end{aligned}$$

(6.137)

It is clear from Eqs. 6.136 and 6.137 that the contribution of the activation overvoltage to the overall voltage loss is significant at low currents.

The Ohmic overvoltage ( $\eta _{ \mathrm{ohm}}$) is due to the resistance to the transport of ions in the electrolyte and to the flow of electrons through the electrodes and current collectors. It is governed by the Ohm’s law:

$$\begin{aligned} \eta _{ \mathrm{ohm}}=iR_{\mathrm{ohm}}. \end{aligned}$$

(6.138)

where $R_{\mathrm{ohm}}$ is the resistance per unit area. The ohmic overvoltage comes into play typically at the middle range of current densities within which the fuel cell is usually designed to operate.

The reactants, i.e., hydrogen and oxygen, in the flow channels have to diffuse through the porous anode and cathode, respectively, to reach the electrode-electrolyte interface where the reaction occurs. Similarly, the product of the reaction, i.e., water vapor, which is formed at the anode electrolyte interface, has to diffuse through the porous anode so as to reach the flow bulk in the anode channel. If the cell is functioning reversibly, the partial pressures of the reactant and the product gas species are the same at the flow bulk in the gas channels and at the TPB where the actual reaction takes place. But when current is drawn from the cell, the partial pressures of the gas species at the TPB differ from their corresponding partial pressures in the bulk due to limitations imposed by the diffusion process (refer to Fig. 6.98). The voltage lost due to this pressure difference between the bulk and the TPB is called the concentration overvoltage.

Fig. 6.98

Schematic representation of the variation of the partial pressures of hydrogen and water vapor through the anode, and oxygen through the cathode

The concentration overvoltage is obtained by subtracting the Nernst voltage obtained by using the partial pressures at the flow bulk and those at the TPB. It is assumed that the pressure loss of hydrogen alone is significant and is responsible for the concentration overvoltage. With this assumption

$$\begin{aligned} \eta _{ \mathrm{conc}}=\dfrac{RT_{\mathrm{M}}}{n_{\mathrm{e}}F}\ln \left( \dfrac{p_{\mathrm{H,b}}}{p_{\mathrm{H,TPB}}}\right) , \end{aligned}$$

(6.139)

which may be further simplified [70] to a form

$$\begin{aligned} \eta _{\mathrm{conc}}=\dfrac{RT_{\mathrm{M}}}{n_{\mathrm{e}}F}\ln \left( \dfrac{i_{ \mathrm{L}}}{i_{\mathrm{L}}-i}\right) =-\dfrac{RT_{\mathrm{M}}}{n_{\mathrm{e}}F}\ln \left( 1-\dfrac{i}{i_{\mathrm{L}}}\right) , \end{aligned}$$

(6.140)

where $i_{\mathrm{L}}$ is the limiting current. The concentration overvoltage is significant only at high currents. From Eq. 6.140, it can be understood that the concentration overvoltage is very less when $i \ll i_{ \mathrm{L}}$. It becomes significantly high when the value of the current approaches the limiting current. Note that Eq. 6.140 is not valid for $i=i_{\mathrm{L}}$.

All these overvoltages are modeled by the RS-field shown in Fig. 6.97. The effort output (for the port with current as the flow input) of the RS-field is given as

$$\begin{aligned} \eta = \dfrac{RT_{\mathrm{M}}}{n_{\mathrm{e}}F}\left( 2\sinh ^{-1}\left( \dfrac{0.5i}{ i_{\mathrm{0,a}}}\right) +2\sinh ^{-1}\left( \dfrac{0.5i}{i_{\mathrm{0,c}}} \right) -\ln \left( 1-\dfrac{i}{i_{\mathrm{L}}}\right) \right) +iR_{\mathrm{ohm}} \end{aligned}$$

(6.141)

and the flow output (for the port with temperature as the effort input), i.e., the entropy flow rate which goes to the heat transfer part of the model, is given as

$$\begin{aligned} \dot{S}_{\mathrm{PL}}=\dfrac{iR}{n_{\mathrm{e}}F}\left( 2\sinh ^{-1}\left( \dfrac{0.5i}{i_{\mathrm{0,a}}}\right) +2\sinh ^{-1}\left( \dfrac{0.5i}{i_{ \mathrm{0,c}}}\right) -\ln \left( 1-\dfrac{i}{i_{\mathrm{L}}}\right) \right) + \dfrac{i^{2}R_{\mathrm{ohm}}}{T_{\mathrm{M}}}.\nonumber \\ \end{aligned}$$

(6.142)

The $0_{T_{\mathrm{M}}}$ junction shown in Fig. 6.97 represents the temperature of the MEA solid. Convection is an important means of heat transfer in an SOFC as the gases flow through the anode and the cathode channels. Due to the ideal gas assumption and the low velocities, the flow in a fuel cell is usually laminar. The bond graph model shown in Fig. 6.97 includes the convective heat transfers between the anode and cathode channel gases, the MEA, and the interconnects. The R-fields, R$_{cv2}$ and R$_{cv4}$, model the convective heat transfers between the gases and the MEA and the R-fields, R$_{cv1}$ and R$_{cv3}$, model the convective heat transfers between the gases and the interconnects denoted by I1 and I2, respectively, in Fig. 6.97. The constitutive relations of the R-field, R$_{cv1}$, is given as [52]

$$\begin{aligned} \dot{S}_{3}&=\dfrac{\lambda _{\mathrm{an}}A_{\mathrm{c}}\left( T_{\mathrm{I1} }-T_{\mathrm{an}}\right) }{T_{\mathrm{an}}} \nonumber \\ \mathrm{and }\;\dot{S}_{4}&=\dfrac{\lambda _{\mathrm{an}}A_{\mathrm{c} }\left( T_{\mathrm{I1}}-T_{\mathrm{an}}\right) }{T_{\mathrm{I1}}}. \end{aligned}$$

(6.143)

The constitutive relations for the other R-field elements defining the convection heat transfer (R$_{cv2}$, R$_{cv3}$ and R$_{cv4}$) are defined in a similar fashion. The thermal capacity of the MEA is represented by the compliance element C$_{M}$ in Fig. 6.97. The constitutive relation of thermal capacity [69] of C$_{M}$ element is given as:

$$\begin{aligned} T_{\mathrm{M}}=T_{\mathrm{M}}^{{0}}\exp \left( \dfrac{S_{\mathrm{M}}-S_{\mathrm{M}}^{{0}}}{m_{\mathrm{M}}c_{\mathrm{M}}}\right) . \end{aligned}$$

(6.144)

The thermal capacitance of the interconnect plates are represented by the two capacitive elements C$_{I1}$ and C$_{I2}$ which are governed by the same constitutive relation given in Eq. 6.144.

The enthalpy of the reaction is given as

$$\begin{aligned} \Delta H=\Delta G+T\Delta S. \end{aligned}$$

(6.145)

where the part $T\Delta S$ is released as heat when the fuel cell operates reversibly. Under irreversible operation (under all realistic circumstances), the change in the Gibbs free energy of the reaction ($\Delta G$) is not completely converted into useful electrical work. Rather, some of it ends up as heat energy. These irreversibilities, which are called overvoltages, give rise to entropy generation and are taken care by the RS-field element in the model. In order to account for the entropy change of the reaction, the entropy flow rate is added to the MEA by means of a modulated source of flow in Fig. 6.97:

$$\begin{aligned} \dot{S}_{\mathrm{r}}=\dfrac{\dot{m}_{\mathrm{H}}^{\mathrm{r}}\left(h_{\mathrm{H} }-\mu _{\mathrm{H}}\right) }{T_{\mathrm{an}}}+\dfrac{\dot{m}_{\mathrm{O}}^{\mathrm{r} }\left( h_{\mathrm{O}}-\mu _{\mathrm{O}}\right) }{T_{\mathrm{ca}}}-\dfrac{\dot{m}_{ \mathrm{W}}^{\mathrm{r}}\left( h_{\mathrm{W}}-\mu _{\mathrm{W}}\right) }{T_{\mathrm{an} }}. \end{aligned}$$

(6.146)

where the specific enthalpies are expressed as follows [5]:

$$\begin{aligned} h=R\left( a_{1}T+a_{2}T^{2}+a_{3}T^{3}+a_{4}T^{4}+a_{5}T^{5}\right) +h_{0}. \end{aligned}$$

(6.147)

The values of the coefficients $a_{1}\cdots a_{6}$ and $h_{0}$ for the different gases are available in thermodynamics handbooks. The source of flow MSf: derivative of $\dot{S}_{\mathrm{r}}$ is modulated with signals $i$ (to calculate $\dot{m}_{\mathrm{H}}^{ \mathrm{r}}$, $\dot{m}_{\mathrm{O}}^{\mathrm{r}}$ and $\dot{m}_{\mathrm{W}}^{\mathrm{r} }$ according to Eq. 10.54), $\mu _{\mathrm{W}}$, $\mu _{\mathrm{H}}$, $\mu _{\mathrm{O}}$, $T_{\mathrm{an}}$ and $T_{\mathrm{ca}}$ (the latter five are calculable from state variables). Note that these modulating signals are not shown in Fig. 6.97 to maintain the visual clarity of the figure.

Unlike the pseudo-bond graphs, the energetic consistency of the true bond graph presented in Fig. 6.97 is apparent. The continuity of energy flows across different domains and across different interfaces is ensured because the effort and the flow variables correspond to the power variables in the corresponding energy domains throughout the bond graph model. All the storage elements in the global model given in Fig. 6.97 are in integral causality. There is no causality violation at any place in the junction structure. This ensures the energy consistency in the model. Moreover, this integrally causalled model does not have algebraic or causal loops, which ensures that this model is well computable.

9.5.4 Static Characteristics

The model needs extensive initialization before simulation. Readers may refer to [70–72] for the parameters used in the simulations and the procedure to calculate initial values of state variables. The fuel utilization (FU) and oxygen utilization (OU) are two of the most important control variables of the fuel cell. Fuel utilization ( $\zeta _{ \mathrm{f}}$) is defined as the ratio of the mass flow rate of the fuel taking part in the reaction to the mass flow rate of the fuel supplied to the cell. Oxygen utilization ( $\zeta _{\mathrm{o}}$) is defined as the ratio of mass flow rate of oxygen consumed by the reaction to the mass flow rate of oxygen supplied to the cell. According to the operational requirement of the SOFC, FU must be maintained constant. Normally, FU of 0.8–0.9 is desired. Usually, a value of 0.8 is chosen for the FU and a value of 0.125 is chosen for the OU.

Figure 6.99 shows the reversible cell voltage as a function of the fuel utilization (FU) with the system pressure as the parameter. The FU is defined as the ratio of fuel mass consumed in the reaction to the supplied fuel mass. From these curves, it is evident that the reversible cell voltage decreases with the increase in the FU and also that increasing system pressure results in increased Nernst voltage. However, this increase is quite small compared to the added complications of operating the cell at high pressure and temperature. Therefore, the cell pressure is usually kept slightly above the atmospheric pressure. For economical viability the cell is operated below 0.9 FU value. Note that a low FU is economically unviable. Some other static characteristic curves of the SOFC can be referred to in [70–72].

Fig. 6.99

Characteristic curves showing the variation of Nernst voltage as a function of FU

Fig. 6.100

The polarisation curve showing the contribution of various voltage losses

The power density and the polarization curves for a cell operating at 1,073 K and 1 bar with undepleted air (zero OU) and FU of 0.03 are shown in Fig. 6.100, where the various internal cell voltage losses are also indicated. For the cell under consideration, it can be seen from Fig. 6.100 that the ohmic and the activation losses are the major losses while the concentration voltage loss is minimum. Concentration losses cause the cell potential to drop to zero sharply with a concave curvature at a current density called the limiting current density [2]. For the cell and the operating conditions chosen in this work, no concave curvature is observed as high ohmic and activation losses cause the cell voltage to drop to zero much before the limiting current density ($i_{ \mathrm{L}}=10$ A/cm$^{{2}}$) is reached.

9.5.5 Fuel Cell Control

In a hydrogen-fed SOFC system, the anode and cathode outlet gases go to an afterburner, where all the remaining hydrogen is combusted, from where they pass through two pre-heaters which are used for heating the inlet hydrogen and air streams. A schematic representation of the full SOFC system is shown in Fig. 6.101. The exhaust gases from the anode and the cathode sides are fed to an afterburner where all the remaining hydrogen is combusted. The exhaust gases from the afterburner first pass through a hydrogen heat exchanger, HX1, where they lose some of their heat energy to the fuel cell inlet hydrogen stream and then they enter another heat exchanger, HX2, where they heat up the fuel cell inlet air and are then released into the atmosphere.

Fig. 6.101

A Schema of the SOFC System with heat exchangers

The afterburner and the heat exchangers are modeled using the pseudo-bond graph approach (the effort and flow variables are $T$ and $\dot{H}$, respectively) as shown in Fig. 6.102. As a usual assumption, the hydraulic storage is neglected in heat-exchanger design [29]. It may be assumed that the hydraulic resistance has been already modeled at the fuel cell exit. As the afterburner and the heat exchanger models are obtained from the first law analysis, it is convenient to represent their dynamics in terms of pseudo-bond graphs. The energy balance equations used in the afterburner modeling are represented in a pseudo-bond graph. To maintain compatibility, the heat exchanger is also modeled as a pseudo-bond graph.

The C$_{AB}$ element in the afterburner section marked in Fig. 6.102 represents the thermal capacitance of the afterburner gases. The two modulated sources of flows to the left of the C$_{AB}$ element give the afterburner inlet enthalpy flow rate ($\dot{H}_{\mathrm{abi}}$) and the enthalpy flow rate due to the hydrogen combustion reaction in the afterburner ($\dot{H}_{\mathrm{r}}$). These flow sources are modulated by the signals of the fuel cell outlet gas species mass flow rates and the cathode gas temperature (which come from the true bond graph model of the fuel cell shown in Fig. 6.97). All the hydrogen gas coming out of the fuel cell is consumed in the instantaneous combustion reaction taking place at the afterburner, which is justified from the fact that the combustion reaction time constant is sufficiently small compared to the other dynamics in the system.

Fig. 6.102

Pseudo-bond graph models of the afterburner and heat exchangers

The coupling element for thermo-fluid (CETF) systems is used to describe convection of multi-component gas mixture (Fig. 6.103). The afterburner outlet enthalpy flow rate, which is also the hot fluid inlet enthalpy flow rate of the hydrogen heat exchanger (HX1), is given by the CETF element. The CETF element is modulated by the afterburner outlet species mass flow rates. The heat exchanger is assumed to be an ideal one in which the pressure and the mass flow rates do not change between the inlet and outlet sides. The thermal capacitances of the hot and cold fluids of the heat exchanger, HX1, are represented by C$_{h}$ and C$_{c}$ elements, respectively, in Fig. 6.102. The capacitance of the solid wall separating the two fluids may be included later in the model. The cold fluid inlet enthalpy flow rate is given by the CETF element which is modulated by the anode inlet mass flow rate signal ($\dot{m}_{\mathrm{an}}^{\mathrm{i}}$), whereas the cold fluid outlet mass flow rate is given by the MSf element which is modulated by the signals of the anode inlet mass flow rate ($\dot{m}_{\mathrm{an}}^{\mathrm{i}}$) and the temperature of the cold fluid. The heat transfer rate between the hot and the cold fluids of the hydrogen heat exchanger (HX1) is modeled by the MR-element (modulated with mass flow rate signals) using the NTU formulation as the constitutive relation. The air heat exchanger (HX2) is also modelled in a similar fashion. The temperatures defined by the cold fluid thermal capacitances (C$_{c}$) of the heat exchangers HX1 and HX2 are given as those of the anode and the cathode inlet temperatures (T$_{ai}$ and T$_{ci}$, respectively) to the SOFC bond graph model shown in Fig. 6.97. The chemical potentials of the gas species at the channel inlets ($\mu _{ \mathrm{H,ai}}$, $\mu _{\mathrm{N,ci}}$ and $\mu _{\mathrm{O,ci}}$) are represented as functions of these temperatures.

Fig. 6.103

Expanded form of CETF element for multi-component gas mixtures

In addition to the constant FU requirement, another requirement for the fuel cell operation is to maintain constant cell temperature despite the changes in the load so as to prevent thermal cracking of the membranes. This is usually achieved by manipulating the cathode inlet mass flow rate (i.e., manipulating the OU). In this work, the constant cell temperature requirement is achieved by means of introducing a PI controller, which manipulates the cathode inlet mass flow rate around a preset value.

9.5.6 SOFC Bond Graph Model with Control System

The true bond graph model discussed before is further improved to include more detailed representation of the overvoltages, inclusion of the conduction and radiation heat transfer effects in the cell, and representation of the flow resistance by isentropic nozzle flow equation rather than a linear flow relation.

The MEA solid is represented by three control volumes; one each for the anode, the cathode, and the electrolyte. The temperatures of the solid anode, cathode, and electrolyte control volumes are represented by the junctions ‘0$_{an,s}$’, ‘0$_{ca,s}$’, and ‘0$_{el}$’, respectively.

Fig. 6.104

True bond graph model of the closed-loop SOFC System

It is assumed that the entropy generated due to the ohmic resistance is added to the solid electrolyte. The ohmic resistance is modeled by the resistive field RS$_{O}$ between the 1$_{i}$ junction and the 0$_{el}$ -junction in Fig. 6.104. The inputs to this field are the current ($i$) and the electrolyte temperature ($T_{ \mathrm{el}}$). The outputs are the overvoltage ($\eta _{\mathrm{ohm}}$) and the entropy flow rate ($\dot{S}_{\mathrm{el}}$), which are calculated as

$$\begin{aligned} \eta _{\mathrm{ohm}}&=iR_{\mathrm{ohm}} \nonumber \\ \mathrm{and }\ \dot{S}_{\mathrm{el}}&=\dfrac{i^{2}R_{\mathrm{ohm}}}{T_{\mathrm{el}}} \end{aligned}$$

(6.148)

The activation and the concentration overvoltages at the anode are modeled by the resistive field element RS$_{\mathrm{A,C}}^{\mathrm{an}}$ between the 1$_{i}$-junction and the 0$_{an,s}$-junction in Fig. 6.104. The inputs to the resistive field are the current and the anode temperature. In addition, this element is modulated with the signals of gas species partial pressures. These signal bonds are not shown in the figure for maintaining the visual clarity. The outputs of the field are the total anodic overvoltage ($\eta _{\mathrm{an}}$) and the entropy generated ($\dot{S}_{ \mathrm{act,conc,an}}$) due to the anodic overvoltages.

Similarly, the activation and the concentration overvoltages at the cathode are modeled by the resistive field element between the 1$_{i}$-junction and the 0$_{ca,s}$-junction in Fig. 6.104, where the 0$_{ca,s}$ -junction represents the common temperature of the cathode solid.

As the SOFC operates at high temperatures, thermal radiation is a significant mode of heat transfer. The radiation heat transfer between the solid anode and the interconnect is modeled by the field element R$_{rd1}$ and that between the solid cathode and the interconnect is modeled in Fig. 6.104 by the field element R$_{rd2}$. The constitutive relations of the field element R$_{rd1}$ are given as

$$\begin{aligned} \dot{S}_{\mathrm{rd1,an,s}}&=\dfrac{\left( \sigma A_{\mathrm{c}}/T_{\mathrm{an,s} }\right) \left( T_{\mathrm{an,s}}^{{4}}-T_{\mathrm{I1}}^{{4}}\right) }{ \left( \left( 1/\varepsilon _{\mathrm{an}}\right) +\left( 1/\varepsilon _{ \mathrm{in}}\right) -1\right) } \end{aligned}$$

(6.149)

$$\begin{aligned} \mathrm{and }\ \dot{S}_{\mathrm{rd1,I1}}&=\dfrac{\left( \sigma A_{\mathrm{c} }/T_{\mathrm{I1}}\right) \left( T_{\mathrm{an,s}}^{{4}}-T_{\mathrm{I1}}^{ {4}}\right) }{\left( \left( 1/\varepsilon _{\mathrm{an}}\right) +\left( 1/\varepsilon _{\mathrm{in}}\right) -1\right) }, \end{aligned}$$

(6.150)

where $\dot{S}_{\mathrm{rd1,an,s}}$ and $\dot{S}_{\mathrm{rd1,I1}}$ are the entropy flow rates at the solid anode and the interconnect sides, respectively. The constitutive relations of the R-field element R$_{rd2}$ are similar.

The mass flow rates through the valve resistances are given by the isentropic nozzle flow equations. The overall mass flow rate of the gas mixture is given by the formulae for the mass flow through an isentropic nozzle as

$$\begin{aligned} \dot{m}=A\left( x\right) \dfrac{p_{\mathrm{u}}}{\sqrt{T_{\mathrm{u}}}}\sqrt{ \dfrac{2\gamma }{R(\gamma -1)}}\sqrt{\left( \dfrac{p_{\mathrm{d}}}{p_{\mathrm{u}} }\right) ^{2/\gamma }-\left( \dfrac{p_{\mathrm{d}}}{p_{\mathrm{u}}}\right) ^{(\gamma +1)/\gamma },} \end{aligned}$$

(6.151)

where the valve areas are given by $A\left( x\right) =Ax$ by assuming linear valve characteristic, i.e., the coefficient of discharge varies linearly with the valve stem displacement.

9.6 SOFC Control

If all the valve displacements are varied proportionally to the current (in the case of a change in load current) then the steady-state operation with the desired values of OU, FU, and channel pressures can be maintained [72]. Note that the reaction mass flow rates are functions of current only. But the temperature of the gases in the channels will not remain constant, i.e., they would vary from their initial steady-state values when the load current changes. Hence, it is necessary to measure both the chamber gas temperatures. The control strategy proposed in this work consists of a primary controller, which simultaneously controls the four valves by varying their valve displacements in order to maintain constant FU and OU.

However, the above-mentioned control action does not ensure constant cell temperature. The usual method followed for controlling the SOFC temperature is the manipulation of the excess air supplied to the cell. For this purpose, a secondary PI temperature controller, which manipulates the air ratio around the value set by the primary controller, is added. The cathode chamber gas temperature is compared with the set point value of the temperature and the objective of the PI controller is to reduce the temperature error signal by manipulating the flow through cathode chamber inlet and outlet valves by means of varying their valve displacements. It is assumed that the economic cost of increased airflow is insignificant with respect to other operational costs.

The outputs of the primary controller are the four valve displacements (see [72] for details). The secondary controller acts only on the cathode inlet and outlet valves by means of manipulating their valve displacements ($x_{ \mathrm{ci}}$ and $x_{\mathrm{co}}$) in order to vary the OU. The modified expression for the cathode inlet valve displacement is given by the sum of the displacements due to the primary controller ($x_{\mathrm{ci}}^{1}$) and the output of the PI controller.

The block diagram of the above control strategy, with both the primary and secondary controllers, is shown in the model given earlier in Fig. 6.104, where the control logic for the blocks FC1, FC2, FC3, and FC4 are given in [72].

The dynamic response (see [70, 71, 73–75] for parameter values and other details) of the open-loop fuel cell to step changes in the load is shown in Fig. 6.105. The load current changes from 100 A to 80 A after 500 s (0 s indicates a reference time in steady-state operation) and again to 90 A after 2,000 s. The open-loop response shows sharp variations in cell temperature and differential pressure (pressure difference between anode and cathode channels). These are detrimental to cell life. Moreover, the FU drops and compromises the cell efficiency. The dynamic response of the closed-loop fuel cell to the same step changes in the load is shown in Fig. 6.106. The OU is varied by the controller so as to bring back the cell temperature to the initial steady-state value. The results show that there is very little change in the partial pressures of the reactants and the product. The voltage increase in this case can be attributed to the decrease in the cell losses due to the lower current density. The pressure difference between the anode and the cathode chambers is small (about 62 Pa). In fact, the pressure difference cannot be made absolutely zero.

Fig. 6.105

Response of the open-loop SOFC System

Fig. 6.106

Response of the closed-loop SOFC System

From these results, it is evident that the temperature and the pressure control requirements are conflicting and some tradeoff between them may be required. The maximum allowable pressure difference value depends on the strength of the membrane support and the age of the fuel cell. In the considered design, the pressure difference, obtained from the simulations (62 Pa) is small. If, in some case, the pressure difference turns out to be large enough then a third controller which resets the temperature set point of the PI controller when the chamber pressure crosses a certain limit may be added in order to retain the pressure difference within the allowable limits. Additionally, the temperature control can also be accomplished by varying the temperature of the input fuel and the air by controlling cold air flow rate to external heat exchangers.

9.7 Proton Exchange Membrane Fuel Cell

The general schematic diagram of a PEMFC is given in Fig. 6.107 . The hydrogen diffuses to the anode catalyst (usually, platinum) and it breaks into protons and electrons (zone 6 in Fig. 6.107). The polymer electrolyte (ionomer) membrane is electrical insulator. Only the protons conduct through the membrane to the cathode. The electrons travel through the external circuit to reach the cathode catalyst (diffusion zone) where oxygen reacts with the electrons and the protons conducted through the electrolyte to form water.

The reaction taking place in a PEMFC is given as

$$\begin{aligned} \mathrm{Anode}&:\mathrm{2H}_{\mathrm{2}}\rightarrow \mathrm{4H}^{ {+}}\mathrm{+4e}^{-} \nonumber \\ \mathrm{Cathode}&:{O}_{\mathrm{2}}\mathrm{+4H}^{{+}} \mathrm{+4e}^{-}\rightarrow \mathrm{2H}_{\mathrm{2}}\mathrm{O+Heat\ \ } \end{aligned}$$

(6.152)

PEMFC is fundamentally different from SOFC due to the type of the reaction and the operating temperature. In a PEMFC, hydrogen ions (protons) diffuse and conduct through the membrane electrode assembly (MEA, which is anode+membrane+cathode) whereas in an SOFC, oxygen ions conduct. Thus, the MEA material and properties are fundamentally different. Moreover, PEMFC is a low temperature cell whereas SOFC is a high temperature cell (requires initial heating to start the reactions). The typical PEMFC operating temperature range is 50–120$^{\circ }$C with nafion membrane and 120–200$^{\circ }$ C with polybenzimidazole (PBI) membrane, the later normally referred to as high-temperature PEMFC. The platinum catalyst is intolerant to impurities like carbon monoxide which is why PBI membranes are being used. PBI membrane allows use of impure hydrogen supply and reduces the operating cost. Depending on the temperature range, the water supplied/produced to/in the cell can be liquid or vapor form.

Fig. 6.107

Schematic diagram of a cut-away of a single PEMFC

Each cell of a PEMFC produces around 1.1 volts (open circuit voltage). Several cells are stacked together to form a PEMFC stack which develops sufficient voltage. Bipolar plates (zone 1 in Fig. 6.107) are used to separate the cells, provide hydrogen and oxygen distribution channels, and are used as terminals to extract the current for the external load.

PEM fuel cells have the highest energy density (see Table 6.3). They also have very fast start-up time (fraction of a second). Thus, they are suitable for use in hybrid electric vehicles, portable power (electronic equipment), and backup power applications.

It may also be noted that a PEMFC has comparable cell efficiency (about 50–70 %) with an SOFC (about 60–70 %). However, PEMFC control requires additional energy. SOFC on the other hand allows recuperation of energy from the high temperature exhaust. When overall efficiency is calculated, it turns out to be about 30–50 % for PEMFC system and about 50–60 % for SOFC system.

9.8 PEMFC Control

Water is not just a product of the reactions in a PEMFC but a necessity to start the reaction. A PEMFC fails to function when the hydration level falls below a minimum threshold and it also fails (efficiency drops) when hydration levels become too high (called flooding of the cell). Thus, water and air management is a key feature of PEMFC control. The membrane must be hydrated to control the rate of protons conducting through the electrolyte. At the same time, water is produced as the product of the reaction. Thus, the water must be evaporated and drained at precisely the same rate as it is produced by the reaction.

If water is removed in excess then the membrane dries up which increases the resistance to flow and eventually may lead to formation of cracks in the membrane. Any crack in the membrane leads to direct contact between hydrogen and oxygen molecules (called a gas "short circuit") and the heat generated by the uncontrolled exothermic reaction would damage the fuel cell. On the other hand, if the water is removed too slowly then the electrodes will flood which would prevent the flow of reactants to the catalyst and stop the reaction. Thus, water content management is the most crucial control parameter in PEMFC operation. To do so, water vapor is usually added to the feed (hydrogen) stream and the air (oxygen) flow rate is changed to remove water vapors. This is a tricky task to achieve properly, because of the limitations of the sensors and actuators. Relative humidity sensors measure the inlet/outlet humidity and it is not possible to obtain the right estimate of the hydration level in the membrane electrode assembly from these measurements. Moreover, actuators such as enthalpy wheel and gas/gas or water/gas membrane humidifier are slow and/or inaccurate. Electroosmotic pumps are found to be somewhat efficient in this regard. Thus, it is not possible to obtain correct measurements (sensor limitation) and at the same time, it is not possible to apply correct corrective action (actuator limitation). This is why, fuzzy-logic controllers are found to be more suitable for PEMFC systems.

Maintaining the right cell temperature is also an important factor in the PEMFC system. The reaction between hydrogen and oxygen is highly exothermic and a large quantity of heat is generated at the reaction site. At the same time, if temperature is not correct, protons do not conduct and reactions do not start. Once the reaction starts, locally high temperatures develop at the reaction sites. However, the same temperature must be maintained throughout the cell. Any difference in temperature across the cell can lead to thermal cracking of the membrane and lead to gas short circuit. This is the same for all kinds of fuel cells. More information on this aspect has been given during discussions on SOFC control. Thus, the cell temperature must be above some minimum value and it must be kept constant to avoid large thermal transients. If there is sudden load change, then the thermal transients must be controlled and the system should transit to new uniform temperature operation as slowly as possible.

The start-up and control circuit for a PEMFC system is shown in Fig. 6.108. The starter battery is used to raise the cell temperature to the operating temperature. The reformer receives some hydrocarbon fuel and supplies hydrogen (may be impure if PBI membrane is used) to the PEMFC stack. The air blower and water supply to the stack perform to tasks: control the cell temperature and control the hydration level.

Fig. 6.108

Schematic diagram of PEMFC stack control and input/output conditioning

The feed water to the PEMFC stack must be pure and free from ions. The purest form of water is called DI (de-ionized) water. DI process at the DI bed removes organic and inorganic ionizable particles from water by a two-phase ion exchange process. Cation and anion resins are used to remove, respectively, the positive and negative ions and replace them with H$^{+}$ ions and OH$^{-}$ (hydroxyl) ions. These ions then combine to form pure water. Resistivity (the level of difficulty for a solution to carry an electrical charge) of the water is used to measure the quality of DI water.

The heat exchanger is used to maintain the cell temperature. The cooling rate depends upon the temperature of the supplied water and the air flow from the air blower.

9.9 PEMFC Bond Graph Model

We will develop a static one-dimensional model of the PEMFC. This static model is based on the models developed in [63] and [60]. The following assumptions are made to develop a simple one-dimensional PEMFC model: the input gases are pure hydrogen and oxygen, the gasses in the anode and cathode channels are in homogeneous distribution (no spatial variation in concentration, pressure, and temperature), the pressure in the cell is constant (entry and exit cell pressures are constant and equal), the gas diffusion is solved in steady state and there are no parasitic reactions. This model is much simpler from the SOFC model developed before in the sense that variations in the cell pressure (and thus, partial pressures), mass (concentration), etc. are not considered. The variation in hydration level occurs very slowly and thus has been neglected in model formulation. The model only considers thermal transients and can be used for obtaining static characteristics of the cell. A model for full transient analysis has to be based on the true bond graph formulation developed for the SOFC model.

Pseudo-power variables are used in the model developed in [63] and [60]. We will keep the model structure almost similar to the cited sources and make minor changes wherever necessary.

9.9.1 Thermochemical Submodel

The hydraulic effort variable pressure ($p$) is considered in bars and the flow variable volume flow rate ($D$) is considered in m$^{3}$/s. Note that the product of effort and flow variables does not yield power due to inconsistency in units. The gas molar flow ($J_{igas}$ considered in mol/s) and molar volume ($V_{m}$ in m$^{3}$/mol) are used to compute the volume flow rate as follows:

$$\begin{aligned} D_{gas}=J_{igas}V_{m}, \end{aligned}$$

(6.153)

where $V_{m}=10^{-5}RT/P_{gas}$ and the factor $10^{-5}$ is used to convert the pressure unit from bar into Pa. Note that the subscript ‘gas’ may refer to H$_{2}$ or O$_{2}$.

The theoretical or reversible cell voltage is given as the potential difference developed between the anode and cathode sides which is given by Nernst equation as

$$\begin{aligned} E=E_{c}-E_{a}=-\dfrac{\Delta G}{n_{\mathrm{e}}F}=\dfrac{\Delta G_{c}-\Delta G_{a}}{n_{\mathrm{e} }F}, \end{aligned}$$

(6.154)

where subscripts c and a, respectively, refer to the cathode and anode, $G$ is the Gibbs free energy, $n_{ \mathrm{e}}$ is number of electrons participating in the reaction per mole of fuel and $F$ is the Faraday’s number. The current generated by the reactions is given as

$$\begin{aligned} i=n_{\mathrm{e}}FJ_{i}. \end{aligned}$$

(6.155)

In SOFC model, the change in Gibb’s free energy has been expressed in terms of specific volumes (in gas mixture because pure gases were not considered), molecular weights and chemical potentials (See Eq. 6.130) and the chemical potentials were evaluated from reference chemical potential (a polynomial function of temperature), temperature and pressure ratio between the current and reference state (See Eq. 6.120). The Gibb’s free energies may be alternatively calculated from the standard reference values as follows:

$$\begin{aligned} \Delta G_{c}=\Delta G_{c}^{0}- \dfrac{RT}{2}\ln \left( P_{O_{2}}\right) \nonumber \\ \Delta G_{a}=\Delta G_{a}^{0}-RT\ln \left( P_{H_{2}}\right) \end{aligned}$$

(6.156)

Note that in the SOFC model, we have used $\ln \left( P_{O_{2}}/P_{O_{2}}^{0}\right)$ and $\ln \left( P_{O_{2}}/P_{H_{2}}^{0}\right)$ which have been simplified here because the pressure is considered in bars and the reference pressure is 1bar. The Gibb’s free energy at the reference state is calculated from the enthalpy and entropy at the reference state as

$$\begin{aligned} \Delta G^{0}=\Delta H^{0}-T\Delta S^{0}, \end{aligned}$$

(6.157)

where the enthalpy difference is the theoretically maximum recoverable energy and $T\Delta S^{0}$ is the reaction heat (loss from theoretical maximum). $\Delta H^{0}$ and $\Delta S^{0}$ represent changes from the initial state and the final state of the reaction. Therefore,

$$\begin{aligned} \Delta H^{0}=\Delta H_{ \mathrm{products}}^{0}-\Delta H_{\mathrm{reactants}}^{0}. \end{aligned}$$

(6.158)

The reference enthalpies and entropies can be calculated from

$$\begin{aligned} \Delta H^{0}\left( T\right) =\Delta H^{0}\left( T_{0}\right) +\int _{T_{0}}^{T}\Delta c_{p}\left( \theta \right) d\theta \end{aligned}$$

(6.159)

and

$$\begin{aligned} \Delta S^{0}\left( T\right) =\Delta S^{0}\left( T_{0}\right) +\int _{T_{0}}^{T}\dfrac{\Delta c_{p}\left( \theta \right) }{\theta }d\theta , \end{aligned}$$

(6.160)

where $c_{p}\left( \theta \right)$ defines the specific heat at constant pressure as a polynomial function of the temperature with virial coefficients as parameters.

Fig. 6.109

Sub-model of thermochemical processes in the anode channel

The bond graph submodel of the thermochemical phenomena in anode channel is given in Fig. 6.109. The model incorporates Eqs. 6.154–6.160 in a block and its effort outputs are summed at a 1-junction to produce the Gibb’s free energy change in the anode side. The 1-junction supplies the information of the rate of moles consumed in the reaction to the thermochemical model block. A TF-element is used to convert the change in Gibb’s free energy into the electric potential according to Nernst equation (Eq. 6.154). The thermochemical reaction block also outputs the reaction heat ($T\Delta S^{0}$) from Eq. 6.157 which produces heating of the anode material. The thermal model supplies the information of local temperature to the thermochemical model block.

The bond graph submodel of the thermochemical phenomena in anode channel is given in the similar manner. Because both oxygen and water vapors are involved in the reactions, there are five power output ports from the thermochemical block model of the cathode side. According to the reaction given in Eq. 6.152, half mole of oxygen is consumed per consumption of one mole of hydrogen to produce one mole of water. Thus, TF-elements with transformer moduli of $\frac{1}{2}$ are used to implement this scaling of variables at the interface to thermochemical model block for cathode channel.

9.9.2 Submodel for Double Layer Capacitor and Overvoltages

In the PEMFC model, the activation, diffusion, and concentration over voltages are modeled in the same way as the SOFC model. In addition, at the interface between electrode and electrolyte, the double layer phenomenon is observed. The double layer phenomenon is modeled by a capacitor which fixes the dynamics of the activation phenomena.

Fig. 6.110

Submodel of double layer capacitor and overvoltages for the anode channel

The submodel for electrochemical field is shown in Fig. 6.110 where the C-element models the double layer capacitor and the RS-element models all over-voltages ($\eta _{a}=\eta _{ \mathrm{act}}+\eta _{\mathrm{conc}}+\eta _{\mathrm{diff}}$) together. The overvoltages produce heat which is taken as an output of the RS-element to the thermal part submodel. The constitutive relation for the RS-element has been already discussed during the development of SOFC model.

9.9.3 Overall Bond Graph Model

The overall model of the PEMFC system are shown in Fig. 6.111. In the bond graph model, the anode chamber thermochemical process submodel is connected to the double-layer capacitor submodel for anode and membrane (electrolyte) boundary. Likewise, thermochemical process submodel for the cathode channel (note the TF-elements with $\frac{1}{2}$ moduli) is connected to the double-layer capacitor submodel for cathode and membrane boundary. The reaction heat from anode and cathode channels and the heat generated due to overvoltages at anode-membrane and cathode-membrane segments (activation, concentration and diffusion losses) are interfaced with the thermal domain submodel given at the right of the figure.

Fig. 6.111

Overall bond graph model of the PEMFC without control components

The potential difference between cathode and anode potentials (taken between two double layer capacitors) minus the ohmic losses ($i^{2}R$ loss) due to internal electrical resistance of the cell is the net potential applied across the external electrical load. This electrical load in turn decides the current drawn from the cell and thus the rate of reaction. The ohmic loss produces heating of the electrolyte which is modeled by another RS-element and the heat generated is interfaced with the thermal domain submodel.

The thermal domain submodel is a simple equivalent network model with lumped capacitances (heat capacity is mass times specific heat) and overall heat transfer coefficients. Heat transfer to the environment through conduction, forced convection (cooling), and radiation takes place at the end plates. The environment temperature ($T_{0}$) is modeled by sources of effort. Each 0-junction in the thermal domain submodel corresponds to a common temperature point. It is assumed that temperatures in local segments (e.g., anode, cathode, electrolyte) are uniform. The temperature information from these 0-junctions are used by RS-elements (see causality of the S bonds) to compute the resistances (overvoltages) as functions of temperature. Likewise, the temperature information are also fed to anode and cathode channel thermochemical block models to compute the components of Gibb’s free energy on the respective sides. The temperature difference between local segments drives heat flow between segments of the cell.